CBSE Class 11 Mathematics Sequences And Series Worksheet Set 03

CBSE Class 11 Mathematics Worksheet - Sequences and Series (2). Students can download these worksheets and practice them. This will help them to get better marks in examinations. Also refer to other worksheets for the same chapter and other subjects too. Use them for better understanding of the subjects.

SECTION A: (1 MARK)

Question. Which term of the G.P \( 18, -12, 8, \dots \) is \( \frac{512}{729} \)?
Answer: \( 9^{\text{th}} \text{ term} \)

Question. Find the sum to infinity: \( 1 + \frac{3}{5} + \frac{9}{25} + \dots \dots \)
Answer: \( 5/2 \)

Question. Insert 4 geometric means between 6 and 192.
Answer: \( 12, 24, 48, 96 \)

Question. Find the G.P whose \( 5^{\text{th}} \) and \( 8^{\text{th}} \) terms are 80 and 640 respectively.
Answer: \( 5, 10, 20, 40, \dots \)

Question. If \( a, b, c \) are in A.P, then find the value of \( \frac{(a-c)^2}{b^2-ac} \).
Answer: 4

Question. Find the sum of the series \( 2^2 + 4^2 + 6^2 + \dots \dots \) to \( n \) terms.
Answer: \( \frac{2n(n+1)(2n+1)}{3} \)

Question. The third term of a G.P. is 3. Find the product of first 5 terms of the G.P.
Answer: 243

Question. What is the \( 20^{\text{th}} \) term of the sequence defined by \( a_n = (n - 1)(n - 2)(3 + n) \)?
Answer: 7866

SECTION B: (4 MARKS)

Question. Find the sum of first \( n \) terms of the series: \( 1 + (1+2) + (1+2+3) + \dots\dots \)
Answer: \( \frac{n(n+1)(n+2)}{6} \)

Question. Find the sum of the first \( n \) terms of the series \( 5 + 7 + 13 + 31 + 85 + \dots \dots \)
Answer: \( \frac{3^n - 1}{2} + 4n \)

Question. The product of first three terms of a G.P. is 1000. If we add 6 to its second term and 7 to its third term, the three terms form an A.P. Find the terms of the G.P.
Answer: \( 5, 10, 20 \text{ or } 20, 10, 5 \)

Question. The first term of a G.P is 1 and the sum of the third and fifth term is 90. Find the common ratio and \( 10^{\text{th}} \) term of the G.P.
Answer: \( r = \pm 3; \pm 729 \)

Question. Find the sum of integers from 1 to 100 that are divisible by 2 or 5.
Answer: 3050

Question. If \( \frac{a^n + b^n}{a^{n-1} + b^{n-1}} \) is the A.M between \( a \) and \( b \), then find the value of \( n \).
Answer: \( n = 1 \)

Question. If the first and \( n^{\text{th}} \) term of a G.P are \( a \) and \( b \) respectively, and if \( P \) is the product of \( n \) terms, prove that \( P^2 = (ab)^n \).
Answer: Let the G.P. be \( a, ar, ar^2, \dots, ar^{n-1} \).
Here, the \( n^{\text{th}} \) term is \( b = ar^{n-1} \).
The product \( P \) of the first \( n \) terms is:
\( P = a \cdot (ar) \cdot (ar^2) \cdots (ar^{n-1}) = a^n r^{1+2+\dots+(n-1)} = a^n r^{\frac{n(n-1)}{2}} \).
Squaring both sides:
\( P^2 = a^{2n} r^{n(n-1)} = (a^2 r^{n-1})^n = (a \cdot ar^{n-1})^n = (ab)^n \).
Hence Proved.

Question. Find the sum to \( n \) terms of the series: \( 3 + 15 + 35 + 63 + \dots \dots \)
Answer: \( \frac{n(4n^2 + 6n - 1)}{3} \)

Question. Find the sum to \( n \) terms of the series \( \frac{1^3}{1} + \frac{1^3 + 2^3}{2} + \frac{1^3 + 2^3 + 3^3}{3} + \dots \dots \)
Answer: \( \frac{n(n+1)(n+2)(3n+5)}{48} \)

Question. The inventor of the chess board suggested a reward of one grain of wheat for the first square, 2 grains for the second, 4 grains for the third and so on, doubling the amount of the grains for subsequent squares. How many grains would have to be given to the inventor? (There are 64 squares in the chess board)
Answer: \( 2^{64} - 1 \)

Question. Three numbers are in A.P and their sum is 15. If 1, 3, 9 be added to them respectively, they form a G.P, find the numbers.
Answer: \( 3, 5, 7 \text{ or } 15, 5, -5 \)

Question. If A.M. and G.M. of two positive numbers \( a \) and \( b \) are 20 and 18 respectively, find the numbers.
Answer: \( 32, 8 \)