Question

An A.P. consists of 'n' terms whose n^th term is 4 and the common difference is 2. If the sum of 'n' terms of A.P. is –14, then find 'n'. Also, find the sum of the first 20 terms.

Answer 1

Given:

n^th term a_n = 4

Common difference d = 2

Sum of n terms S_n = –14

Step-wise calculation:

1. Express the n^th term: a + (n – 1)d = 4.

With d = 2: a + 2(n – 1) = 4

⇒ a + 2n – 2 = 4

⇒ a + 2n = 6

So a = 6 – 2n.

2. Use the sum formula `S_n = n/2 (a + a_n)`. 

Here a_n = 4 and S_n = –14: `n/2 (a + 4) = -14`.

3. Substitute a = 6 – 2n into the sum:

`(n/2)((6 − 2n) + 4) = -14 (n/2)(10 − 2n)`

= –14 n(5 – n) 

= –14 

⇒ n² – 5n – 14 = 0 

⇒ (n – 7)(n + 2) = 0

4. n must be a positive integer, so n = 7 (reject n = –2). 

Using n = 7, find a: a = 6 – 2n

= 6 – 14

= –8

5. Sum of first 20 terms `S_20 = 20/2 [2a + (20 - 1)d]`: 

S₂₀ = 10[2(–8) + 19 × 2] 

= 10[–16 + 38]

= 10 × 22

= 220