Question

sum of first 7 terms of AP is 49 and that of 17 terms is 289. Find sum of first n terms​

Answer 1

Answer:

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. Here S7 = 49, S17 = 289 Let 'a' be the first term and 'd' the common difference.

Answer 2

Given:-

• The sum of the first 7 term of an ap is 49 and 17 term is 289.

To find:-

• Find the sum of first n term...?

Solutions:-

• The sum of 7 term of an Ap is 49
• The sum of 17 term of Ap is 289.

we know,

The sum of 7 term of an Ap is 49.

=> Sn = n/2 [2a + (n - 1)d]

=> S7 = 7/2 [2a + (7 - 1)d]

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

=> 49 = 7/2 × 2[a + 3d]

=> 49 = 7 [a + 3d]

=> 49/7 = a + 3d

=> 7 = a + 3d..................(i).

The sum of 17 term of Ap is 289.

=> Sn = n/2 [2a + (n - 1)d]

=> S17 = 17/2 [2a + (17 - 1)d]

=> 289 = 17/2 [2a + 16d]

=> 289 = 17/2 × 2[a + 8d]

=> 289 = 17 [a + 8d]

=> 289/17 = a + 8d

=> 17 = a + 8d..................(ii).

Now, Subtracting Eq. (ii) and (i) we get,

${a} \: + \: {8d} \: \: = \: \: {17} \\ {a} \: + \: {3d} \: \: = \: \: {7} \\ \underline{ - \: \: \: \: \: \: \: \: - \: \: \: \: \: \: \: \: = \: \: \: \: \: \: - \: \: \: \: \: \: \: \: \: } \\ \: \: \: \: \: \: \: \: {5d} \: \: \: \: \: \: \: \: = \: \: \: {10}$

=> d = 10/5

=> d = 2

Now, putting the value of d in Eq. (i).

=> a + 3d = 7

=> a + 3 × 2 = 7

=> a + 6 = 7

=> a = 7 - 6

=> a = 1

Thus, a = 1, d = 2, n = ?

=> Sn = n/2 [2a + (n - 1)d]

=> Sn = n/2 [2 × 1 + (n - 1) (2)]

=> Sn = n/2 [2 + 2n - 2]

=> Sn = n/2[2n]

=> Sn = n/2 × 2n

=> Sn = 2n²/2

=> Sn = n²

Hence, the sum of first n term is n².