JEE Mathematics Sequence and Series MCQs Set 44

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Multiple Choice Questions

Question 1: Statement I: a sequence is called an arithmetic progression if the difference of a term and the previous term is always the same. Statement II: the arithmetic mean A of any two numbers a and b is (1/2)(a + b) such that a, A, b are in A.P.; for n positive numbers the A.M. is (a1 + a2 + ... + an)/n. Which statements are correct?
(a) Both are true
(b) Only I is true
(c) Only II is true
(d) Both are false
Show Answer & Explanation

Answer: (a) Both are true

Explanation:
1. Statement I is the definition of an \( A.P \).
2. Statement II is the definition of arithmetic mean, with \( A.P \). check \( (A - a = b - A) \).

Teacher's Note:
1. Express the given condition through the first term and the common difference or ratio, then solve for the unknowns.
2. Remember the standard facts: AP has a constant difference, GP a constant ratio, and HP has reciprocals in AP.

Question 2: Statement I: if a is the first term and d the common difference of an A.P., its nth term is a(n) = a - (n - 1)d. Statement II: the sum S(n) of n terms of an A.P. with first term a and common difference d is S(n) = (n/2){2a + (n - 1)d}. Which statements are correct?
(a) Only II is true
(b) Only I is true
(c) Both are true
(d) Both are false
Show Answer & Explanation

Answer: (a) Only II is true

Explanation:
1. The nth term is \( a + (n - 1)d \), so Statement I is false (wrong sign).
2. Statement II is the correct sum formula.

Teacher's Note:
1. Express the given condition through the first term and the common difference or ratio, then solve for the unknowns.
2. Remember the standard facts: AP has a constant difference, GP a constant ratio, and HP has reciprocals in AP.

Question 3: Statement I: the nth term of a G.P. with first term a and common ratio r is \( a(n) = a r^{n-1} \). Statement II: the geometric mean of a and b is \( (ab)^{1/3} \). Which statements are correct?
(a) Only I is true
(b) Only II is true
(c) Both are true
(d) Both are false
Show Answer & Explanation

Answer: (a) Only I is true

Explanation:
1. Statement I is the standard formula.
2. The geometric mean of two numbers is the square root of ab, i.e. \( (ab)^{1/2} \), so II is false.

Teacher's Note:
1. Express the given condition through the first term and the common difference or ratio, then solve for the unknowns.
2. Remember the standard facts: AP has a constant difference, GP a constant ratio, and HP has reciprocals in AP.

Question 4: I: if each term of a G.P. is raised to the same power, the resulting sequence also forms a G.P. II: the 25th term of the sequence 4, 9, 14, 19, ... is 124. Which are true?
(a) Both are true
(b) Both are false
(c) Only I is true
(d) Only II is true
Show Answer & Explanation

Answer: (a) Both are true

Explanation:
1. Raising to the power \( k \) multiplies the ratio to \( r^k \), so I is true.
2. 25th term = \( 4 + 24 x 5 = 124 \), so II is true.

Teacher's Note:
1. Express the given condition through the first term and the common difference or ratio, then solve for the unknowns.
2. Remember the standard facts: AP has a constant difference, GP a constant ratio, and HP has reciprocals in AP.

Question 5: I: the 18th term of the sequence 72, 70, 68, 66, ... is 40. II: the 4th term of the sequence 8 - 6i, 7 - 4i, 6 - 2i, ... is purely real. Which are true?
(a) Only II is true
(b) Only I is true
(c) Both are true
(d) Both are false
Show Answer & Explanation

Answer: (a) Only II is true

Explanation:
1. I: \( d = -2 \), so the 18th term is \( 72 - 34 = 38 \), not 40; I is false.
2. II: \( d = -1 + 2i \), 4th term = \( (6 - 2i) + (-1 + 2i) = 5 \), which is real; II is true.

Teacher's Note:
1. Express the given condition through the first term and the common difference or ratio, then solve for the unknowns.
2. Remember the standard facts: AP has a constant difference, GP a constant ratio, and HP has reciprocals in AP.

Question 6: I: there are 37 terms in the sequence 3, 6, 9, 12, ..., 111. II: the general term of the sequence 9, 12, 15, 18, ... is 3n + 8. Which are true?
(a) Only I is true
(b) Only II is true
(c) Both are true
(d) Both are false
Show Answer & Explanation

Answer: (a) Only I is true

Explanation:
1. I: \( 111 = 3 x 37 \), so there are 37 terms, true.
2. II: the general term is \( 9 + 3(n - 1) = 3n + 6 \), not \( 3n + 8 \), so II is false.

Teacher's Note:
1. Express the given condition through the first term and the common difference or ratio, then solve for the unknowns.
2. Remember the standard facts: AP has a constant difference, GP a constant ratio, and HP has reciprocals in AP.

Question 7: Match: A. sum of 20 terms of the A.P. 1, 4, 7, 10, ...; B. sum of the series 5 + 13 + 21 + ... + 181; C. sum of all three-digit natural numbers divisible by 7; D. sum of all natural numbers between 250 and 1000 which are exactly divisible by 3; with 1. 70336, 2. 156375, 3. 2139, 4. 590. The correct pairing is
(a) A-4, B-3, C-1, D-2
(b) A-4, B-1, C-3, D-2
(c) A-2, B-3, C-1, D-4
(d) A-3, B-4, C-2, D-1
Show Answer & Explanation

Answer: (a) A-4, B-3, C-1, D-2

Explanation:
1. A: \( 10(2 + 57) = 590 \).
2. \( B \): 23 terms, \( 23(5 + 181)/2 = 2139 \).
3. \( C \): 105 to 994, 128 terms, \( 64 x 1099 = 70336 \).
4. \( D \): 252 to 999, 250 terms, \( 125 x 1251 = 156375 \).

Teacher's Note:
1. Express the given condition through the first term and the common difference or ratio, then solve for the unknowns.
2. Remember the standard facts: AP has a constant difference, GP a constant ratio, and HP has reciprocals in AP.

Question 8: Match: A. sum of 7 terms of the G.P. 3, 6, 12, ...; B. sum of 10 terms of the G.P. 1, 1/2, 1/4, ...; C. sum of the series 2 + 6 + 18 + ... + 4374; D. sum to n terms of the series 11 + 103 + 1005 + ...; with 1. \( (10/9)(10^n - 1) + n^2, 2 \). 1023/512, 3. 381, 4. 6560. The correct pairing is
(a) A-3, B-2, C-4, D-1
(b) A-3, B-4, C-2, D-1
(c) A-1, B-2, C-3, D-4
(d) A-2, B-3, C-4, D-1
Show Answer & Explanation

Answer: (a) A-3, B-2, C-4, D-1

Explanation:
1. A: \( 3(2^7 - 1) = 381 \).
2. \( B \): \( 2(1 - 2^{-10}) = 1023/512 \).
3. \( C \): \( 4374 = 2 x 3^7 \) so 8 terms, sum = \( 2(3^8 - 1)/2 = 6560 \).
4. \( D \): terms are \( 10^k + (2k - 1) \), so the sum is \( (10/9)(10^n - 1) + n^2 \).

Teacher's Note:
1. Express the given condition through the first term and the common difference or ratio, then solve for the unknowns.
2. Remember the standard facts: AP has a constant difference, GP a constant ratio, and HP has reciprocals in AP.

Question 9: If the sequence is defined by a(n) = n(n + 2), match a1, a2, a3, a4, a5 with 1. 35, 2. 24, 3. 8, 4. 3, 5. 15. The correct pairing is
(a) a1-4, a2-2, a3-5, a4-3, a5-1
(b) a1-1, a2-3, a3-2, a4-5, a5-4
(c) a1-3, a2-4, a3-5, a4-1, a5-2
(d) a1-4, a2-3, a3-5, a4-2, a5-1
Show Answer & Explanation

Answer: (d) a1-4, a2-3, a3-5, a4-2, a5-1

Explanation:
1. \( a1 = 3, a2 = 8, a3 = 15, a4 = 24, a5 = 35 \), which pair as 4, 3, 5, 2, 1.

Teacher's Note:
1. Express the given condition through the first term and the common difference or ratio, then solve for the unknowns.
2. Remember the standard facts: AP has a constant difference, GP a constant ratio, and HP has reciprocals in AP.

Question 10: If the nth term of a sequence is a(n) = (2n - 3)/6, match a1 to a5 with 1. 1/6, 2. 1/2, 3. 5/6, 4. -1/6, 5. 7/6. The correct pairing is
(a) a1-4, a2-1, a3-2, a4-3, a5-5
(b) a1-4, a2-1, a3-3, a4-2, a5-5
(c) a1-5, a2-3, a3-2, a4-1, a5-4
(d) a1-4, a2-3, a3-3, a4-1, a5-5
Show Answer & Explanation

Answer: (a) a1-4, a2-1, a3-2, a4-3, a5-5

Explanation:
1. \( a1 = -1/6, a2 = 1/6, a3 = 3/6 = 1/2, a4 = 5/6, a5 = 7/6 \).

Teacher's Note:
1. Express the given condition through the first term and the common difference or ratio, then solve for the unknowns.
2. Remember the standard facts: AP has a constant difference, GP a constant ratio, and HP has reciprocals in AP.

Question 11: The Fibonacci sequence is defined by \(1=a_1=a_2\) and \(a_n=a_{n-1}+a_{n-2}\), \(n>2\). The value of \(\dfrac{a_{n+1}}{a_n}\) for \(n=2\) is
(a) 1
(b) 2
(c) 3
(d) 4
Show Answer & Explanation

Answer: (b) 2

Explanation:
1. \(a_3=a_2+a_1=2\), so \(\dfrac{a_3}{a_2}=2\).

Teacher's Note:
1. Express the given condition through the first term and the common difference or ratio, then solve for the unknowns.
2. Remember the standard facts: AP has a constant difference, GP a constant ratio, and HP has reciprocals in AP.

Question 12: If the sum of a certain number of terms of the A.P. 25, 22, 19, ... is 116, then the last term is
(a) 0
(b) 2
(c) 6
(d) 4
Show Answer & Explanation

Answer: (d) 4

Explanation:
1. \(\dfrac n2[50-3(n-1)]=116\) gives \(3n^2-53n+232=0\), so \(n=8\) (the other root \(29/3\) is not an integer).
2. Last term \(=25-3\cdot7=4\).

Teacher's Note:
1. Express the given condition through the first term and the common difference or ratio, then solve for the unknowns.
2. Remember the standard facts: AP has a constant difference, GP a constant ratio, and HP has reciprocals in AP.

Question 13: If \(\dfrac{a^n+b^n}{a^{n-1}+b^{n-1}}\) is the A.M. between \(a\) and \(b\), then the value of \(n\) is
(a) 2
(b) 1
(c) 3
(d) 4
Show Answer & Explanation

Answer: (b) 1

Explanation:
1. \(2(a^n+b^n)=(a+b)(a^{n-1}+b^{n-1})\) simplifies to \((a-b)(a^{n-1}-b^{n-1})=0\), so \(n=1\).

Teacher's Note:
1. Express the given condition through the first term and the common difference or ratio, then solve for the unknowns.
2. Remember the standard facts: AP has a constant difference, GP a constant ratio, and HP has reciprocals in AP.

Question 14: The difference between any two consecutive interior angles of a polygon is 5 degrees. If the smallest angle is 120 degrees, the number of the sides of the polygon is
(a) 6
(b) 8
(c) 9
(d) 5
Show Answer & Explanation

Answer: (c) 9

Explanation:
1. \((n-2)180=\dfrac n2[240+5(n-1)]\) gives \(n^2-25n+144=0\), so \(n=9\) or \(16\).
2. For \( n = 16 \) the largest angle is 195 degrees, which exceeds 180, so \( n = 9 \).

Teacher's Note:
1. Express the given condition through the first term and the common difference or ratio, then solve for the unknowns.
2. Remember the standard facts: AP has a constant difference, GP a constant ratio, and HP has reciprocals in AP.

Question 15: How many terms of the G.P. \(3,3^2,3^3,\ldots\) are needed to give the sum 120?
(a) 4
(b) 3
(c) 5
(d) 6
Show Answer & Explanation

Answer: (a) 4

Explanation:
1. \(\dfrac{3(3^n-1)}{2}=120\) gives \(3^n=81\), so \(n=4\).

Teacher's Note:
1. Express the given condition through the first term and the common difference or ratio, then solve for the unknowns.
2. Remember the standard facts: AP has a constant difference, GP a constant ratio, and HP has reciprocals in AP.

Question 16: A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying odd places, then the common ratio is
(a) 5
(b) 1
(c) 3
(d) 4
Show Answer & Explanation

Answer: (d) 4

Explanation:
1. Total = odd-place sum + even-place sum, so even-place sum = \( 4 x \) odd-place sum.
2. Even-place terms are \(r\) times the odd-place terms, so \(r=4\).

Teacher's Note:
1. Express the given condition through the first term and the common difference or ratio, then solve for the unknowns.
2. Remember the standard facts: AP has a constant difference, GP a constant ratio, and HP has reciprocals in AP.

Question 17: How many terms of the geometric series \(1+4+16+64+\cdots\) will make the sum 5461?
(a) 3
(b) 4
(c) 7
(d) 5
Show Answer & Explanation

Answer: (c) 7

Explanation:
1. \(\dfrac{4^n-1}3=5461\) gives \(4^n=16384=4^7\), so \(n=7\).

Teacher's Note:
1. Express the given condition through the first term and the common difference or ratio, then solve for the unknowns.
2. Remember the standard facts: AP has a constant difference, GP a constant ratio, and HP has reciprocals in AP.

Question 18: Assertion: for x = 1 or -1, the numbers -2/7, x, -7/2 are in G.P. Reason: three numbers a, b, c are in G.P. if \( b^2 \) = ac. Which is correct?
(a) Both are true but Reason does not explain Assertion
(b) Assertion is true, Reason is false
(c) Assertion is false, Reason is true
(d) Both are true and Reason explains Assertion
Show Answer & Explanation

Answer: (d) Both are true and Reason explains Assertion

Explanation:
1. \( x^2 = (-2/7)(-7/2) = 1 \), so \( x = 1 \) or \( -1 \).
2. This uses exactly the condition \( b^2 \) = ac of the reason.

Teacher's Note:
1. Express the given condition through the first term and the common difference or ratio, then solve for the unknowns.
2. Remember the standard facts: AP has a constant difference, GP a constant ratio, and HP has reciprocals in AP.

Question 19: Assertion: the sum to n terms of the geometric progression \( x^3, x^5, x^7 \), ... (x not equal to +-1) is \( x^3(1 - x^{2n})/(1 - x^2) \). Reason: if a is the first term and r the common ratio of a G.P., the sum to n terms is \( S = a(r^n - 1)/(r - 1) \) or \( a(1 - r^n)/(1 - r) \) if r is not 1. Which is correct?
(a) Both are true but Reason does not explain Assertion
(b) Assertion is true, Reason is false
(c) Assertion is false, Reason is true
(d) Both are true and Reason explains Assertion
Show Answer & Explanation

Answer: (d) Both are true and Reason explains Assertion

Explanation:
1. Here \( a = x^3 \) and \( r = x^2 \), so \( S = x^3(1 - x^{2n})/(1 - x^2) \).
2. This is the reason's formula applied.

Teacher's Note:
1. Express the given condition through the first term and the common difference or ratio, then solve for the unknowns.
2. Remember the standard facts: AP has a constant difference, GP a constant ratio, and HP has reciprocals in AP.

Question 20: Assertion: if a, b, c are in A.P., then b + c, c + a, a + b are in A.P. Reason: if a, b, c are in A.P., then \( 10^a, 10^b, 10^c \) are in G.P. Which is correct?
(a) Both are true and Reason explains Assertion
(b) Assertion is true, Reason is false
(c) Assertion is false, Reason is true
(d) Both are true but Reason does not explain Assertion
Show Answer & Explanation

Answer: (d) Both are true but Reason does not explain Assertion

Explanation:
1. Subtracting from the sum: the terms are \( S - a, S - b, S - c \), which are in \( A.P \). when a, \( b, c \) are.
2. \( 10^b / 10^a = 10^{b - a} = 10^{c - b} \) is constant, so the reason is true; but it does not explain the assertion.

Teacher's Note:
1. Express the given condition through the first term and the common difference or ratio, then solve for the unknowns.
2. Remember the standard facts: AP has a constant difference, GP a constant ratio, and HP has reciprocals in AP.

Question 21: Assertion: if \(\dfrac23,k,\dfrac58\) are in A.P., then the value of \(k\) is \(\dfrac{31}{48}\). Reason: three numbers a, b, c are in A.P. iff 2b = a + c. Which is correct?
(a) Both are true but Reason does not explain Assertion
(b) Assertion is true, Reason is false
(c) Assertion is false, Reason is true
(d) Both are true and Reason explains Assertion
Show Answer & Explanation

Answer: (d) Both are true and Reason explains Assertion

Explanation:
1. \(2k=\dfrac23+\dfrac58=\dfrac{31}{24}\), so \(k=\dfrac{31}{48}\).

Teacher's Note:
1. Express the given condition through the first term and the common difference or ratio, then solve for the unknowns.
2. Remember the standard facts: AP has a constant difference, GP a constant ratio, and HP has reciprocals in AP.

Question 22: Assertion: the 20th term of the series 2 x 4 + 4 x 6 + 6 x 8 + ... + n terms is 1680. Reason: if the sum of three numbers in A.P. is 24 and their product is 440, the numbers are 5, 8, 11 or 11, 8, 5. Which is correct?
(a) Both are true and Reason explains Assertion
(b) Assertion is true, Reason is false
(c) Assertion is false, Reason is true
(d) Both are true but Reason does not explain Assertion
Show Answer & Explanation

Answer: (d) Both are true but Reason does not explain Assertion

Explanation:
1. The k-th term is \( 2k(2k + 2) \); for \( k = 20 \) it is \( 40 x 42 = 1680 \).
2. The middle number is 8 and \( (8 - d)(8)(8 + d) = 440 \) gives \( d = \pm 3 \). Both are true but independent.

Teacher's Note:
1. Express the given condition through the first term and the common difference or ratio, then solve for the unknowns.
2. Remember the standard facts: AP has a constant difference, GP a constant ratio, and HP has reciprocals in AP.

Question 23: Assertion: the sum of n terms of the A.P. whose kth term is 5k + 1 is n(5n + 7)/2. Reason: the sum of all natural numbers lying between 100 and 1000 which are multiples of 5 is 980. Which is correct?
(a) Both are true and Reason explains Assertion
(b) Assertion is true, Reason is false
(c) Both are true but Reason does not explain Assertion
(d) Assertion is false, Reason is true
Show Answer & Explanation

Answer: (b) Assertion is true, Reason is false

Explanation:
1. Sum = \( 5n(n + 1)/2 + n = n(5n + 7)/2 \), so the assertion is true.
2. Multiples of 5 from 105 to 995 give 179 terms with sum \( 179 x 550 = 98450 \), not 980, so the reason is false.

Teacher's Note:
1. Express the given condition through the first term and the common difference or ratio, then solve for the unknowns.
2. Remember the standard facts: AP has a constant difference, GP a constant ratio, and HP has reciprocals in AP.

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