CBSE Class 10 Mathematics Arithmetic Progression Notes Set 07

Sequences, Series and Progressions

Sequences

Sequences are sets of numbers arranged according to a specific rule. The set of numbers should have a definite, logical rule according to which they are arranged. It need not be a mathematical formula, but it should be logical. Such a set of numbers is called a sequence.

Series

A series is a sequence of numbers added by '+' signs, for example 2 + 4 + 6 + 8 + ... and 3 + 5 + 7 + ... . The term 'series' is closely related to the sum of a sequence of numbers; the numbers in a series are separated by plus signs, whereas the numbers in a sequence are separated by commas. A series is always based on a sequence.

Progressions

Progressions are sequences arranged according to a definite rule. The difference between a progression and a sequence is that a progression has a specific formula to calculate its nth term, whereas a sequence can be based on a logical rule such as 'the prime numbers', which has no formula for the nth term. All progressions are sequences, but not all sequences are progressions (for example the prime numbers). The three general types of progressions are the Arithmetic Progression (A.P.), the Geometric Progression (G.P.) and the Harmonic Progression (H.P.). In this chapter we deal only with the A.P.

Arithmetic Progression

A sequence is called an Arithmetic Progression (A.P.) if the difference of a term and the previous term is always the same, that is \( d = t_{n+1} - t_n = \) constant for all \( n \in \mathbb{N} \). The constant difference, generally denoted by d, is called the common difference.

General form of an A.P. If the first term is a and the common difference is d, then \( a, a + d, a + 2d, a + 3d, a + 4d, \dots \) forms an A.P.

nth term of an A.P. For the A.P. \( a, a + d, a + 2d, \dots \): first term \( a_1 = a + 0\cdot d \), second term \( a_2 = a + 1\cdot d \), third term \( a_3 = a + 2d \), and so on. Hence the nth term is

\[ a_n = a + (n - 1)d. \]

mth term of an A.P. from the end. Let a be the first term and d the common difference of an A.P. having n terms. Then the mth term from the end is the \( (n - m + 1) \)th term from the beginning, that is the \( \{n - (m - 1)\} \)th term from the beginning.

Solved Examples

Example 1. If x + 1, 3x and 4x + 2 are in A.P., find the value of x.
Answer:
Since they are in A.P., twice the middle term is the sum of the extremes: \( 2(3x) = (x + 1) + (4x + 2) \Rightarrow 6x = 5x + 3 \Rightarrow x = 3 \).

Example 2 (a). Find the common difference of the A.P. 1, 4, 7, 10, 13, 16, ...
Answer:
\( d = 4 - 1 = 7 - 4 = 10 - 7 = 13 - 10 = 16 - 13 = 3 \) (constant). The common difference is 3.

Example 2 (b). Find the A.P. whose first term is 10 and common difference is 5.
Answer:
First term a = 10 and common difference d = 5. The A.P. is 10, 15, 20, 25, 30, ...

Example 3. Determine the A.P. whose third term is 16 and the difference of the 5th term from the 7th term is 12.
Answer:
\( a_3 = a + 2d = 16 \) ... (i) and \( a_7 - a_5 = 12 \) ... (ii).
From (ii): \( (a + 6d) - (a + 4d) = 12 \Rightarrow 2d = 12 \Rightarrow d = 6 \). From (i): \( a = 16 - 12 = 4 \).
The A.P. is 4, 10, 16, 22, 28, ...

Example 4. Which term of the sequence 72, 70, 68, 66, ... is 40?
Answer:
a = 72, \( d = 70 - 72 = -2 \). \( a_n = a + (n - 1)d \Rightarrow 40 = 72 + (n - 1)(-2) \Rightarrow -32 = -2n + 2 \Rightarrow n = 17 \).
The 17th term is 40.

Example 5. Is 184 a term of the sequence 3, 7, 11, ...?
Answer:
a = 3, d = 4. \( 184 = 3 + (n - 1)4 \Rightarrow 181 = 4n - 4 \Rightarrow n = \dfrac{185}{4} \).
Since n is not a natural number, 184 is not a term of the given sequence.

Example 6. Which term of the sequence 20, \( 19\tfrac{1}{4} \), \( 18\tfrac{1}{2} \), \( 17\tfrac{3}{4} \), ... is the first negative term?
Answer:
a = 20, \( d = 19\tfrac{1}{4} - 20 = -\tfrac{3}{4} \). Let \( a_n < 0 \): \( 20 + (n - 1)\left(-\tfrac{3}{4}\right) < 0 \Rightarrow \dfrac{83}{4} - \dfrac{3n}{4} < 0 \Rightarrow n > \dfrac{83}{3} = 27\tfrac{2}{3} \).
Since 28 is the natural number just greater than \( 27\tfrac{2}{3} \), the first negative term is the 28th term.

Example 7. If the pth, qth and rth terms of an A.P. are a, b, c respectively, show that \( a(q - r) + b(r - p) + c(p - q) = 0 \).
Answer:
Let A be the first term and D the common difference: \( a = A + (p - 1)D \), \( b = A + (q - 1)D \), \( c = A + (r - 1)D \).
L.H.S. \( = \{A + (p - 1)D\}(q - r) + \{A + (q - 1)D\}(r - p) + \{A + (r - 1)D\}(p - q) \).
The coefficient of A is \( (q - r) + (r - p) + (p - q) = 0 \), and the coefficient of D is \( (p - 1)(q - r) + (q - 1)(r - p) + (r - 1)(p - q) = 0 \) (the terms in p, q, r cancel in pairs). So L.H.S. = 0 = R.H.S.

Example 8. If m times the mth term of an A.P. is equal to n times its nth term, show that the (m + n)th term of the A.P. is zero.
Answer:
Let A be the first term and D the common difference. \( m\,a_m = n\,a_n \Rightarrow m[A + (m - 1)D] = n[A + (n - 1)D] \).
\( \Rightarrow A(m - n) + D[m(m - 1) - n(n - 1)] = 0 \Rightarrow (m - n)[A + (m + n - 1)D] = 0 \). Since \( m \ne n \), \( A + (m + n - 1)D = 0 \), that is \( a_{m+n} = 0 \).

Example 9. If the pth term of an A.P. is q and the qth term is p, prove that its nth term is (p + q - n).
Answer:
\( A + (p - 1)D = q \) ... (i) and \( A + (q - 1)D = p \) ... (ii). Subtracting, \( (p - q)D = q - p \Rightarrow D = -1 \). Then \( A = q + (p - 1) = p + q - 1 \).
\( a_n = A + (n - 1)D = (p + q - 1) - (n - 1) = p + q - n \).

Example 10. If the mth term of an A.P. is \( \dfrac{1}{n} \) and the nth term is \( \dfrac{1}{m} \), show that its (mn)th term is 1.
Answer:
\( A + (m - 1)D = \dfrac{1}{n} \) ... (i), \( A + (n - 1)D = \dfrac{1}{m} \) ... (ii). Subtracting, \( (m - n)D = \dfrac{m - n}{mn} \Rightarrow D = \dfrac{1}{mn} \). Then \( A = \dfrac{1}{n} - \dfrac{m - 1}{mn} = \dfrac{1}{mn} \).
\( a_{mn} = A + (mn - 1)D = \dfrac{1}{mn} + \dfrac{mn - 1}{mn} = 1 \).

Example 11. Find the 20th term from the end of the A.P. 3, 7, 11, ..., 407.
Answer:
a = 3, d = 4, \( a_n = 407 \Rightarrow 3 + (n - 1)4 = 407 \Rightarrow n = 102 \).
The 20th term from the end is the \( (102 - 20 + 1) = 83 \)rd term from the beginning: \( a_{83} = 3 + 82 \times 4 = 331 \).

Check Your Progress

1. In an A.P. the first term is 2, the last term is 29 and the sum is 155. Find the common difference.
Answer:
\( S_n = \dfrac{n}{2}(a + l) \Rightarrow 155 = \dfrac{n}{2}(31) \Rightarrow n = 10 \). Then \( 29 = 2 + 9d \Rightarrow d = 3 \). d = 3

2. What is the value of k for which the numbers k, k - 2, 3k are in A.P.?
Answer:
\( 2(k - 2) = k + 3k \Rightarrow 2k - 4 = 4k \Rightarrow \) k = -2

3. Find the 15th term of the A.P. x - 7, x - 2, x + 3, ...
Answer:
d = 5, so \( a_{15} = (x - 7) + 14 \times 5 = \) x + 63

4. If the 8th and 12th terms of an A.P. are 32 and 52 respectively, find the 20th term.
Answer:
\( a + 7d = 32 \) and \( a + 11d = 52 \Rightarrow d = 5, a = -3 \). \( a_{20} = -3 + 19 \times 5 = \) 92

5. Find whether -150 is a term of the A.P. 17, 12, 7, 2, ...
Answer:
a = 17, d = -5. \( 17 - 5(n - 1) = -150 \Rightarrow n - 1 = 33.4 \), not a whole number. No, -150 is not a term.