If the sum of first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of first n terms.

Solution:

Given that,

S₇ = 49

S₁₇ = 289

We know the sum of n terms;

S_n = n /2 [2a + (n – 1) d ]

Therefore,

S₇ = 7 /2 [2a + (n – 1) d ]

S₇ = 7/2 [2a + (7 – 1) d ]

49 = 7/2 [2a + 6 d ]

7 = (a + 3d)

a + 3d = 7 …………………………………. (i)

Similarly,

S₁₇ = 17/2 [2a + (17 – 1) d ]

289 = 17/2 (2a + 16 d )

17 = ( a + 8d )

a + 8d = 17 ………………………………. (ii)

Subtracting equation (i) from equation (ii),

5 d = 10

d = 2

From equation (i),

a + 3(2) = 7

a + 6 = 7

a = 1

Hence,

S_n = n /2[2a + (n – 1) d ]

= n /2[2(1) + (n – 1) × 2]

= n /2(2 + 2n – 2)

= n /2(2n)

= n²