Question

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​

Answer 1

» Sum of first 7 terms of an AP is 49 and that of 17 terms is 289.

$S_{n}$ = $\dfrac{n}{2}$ [2a + (n - 1)d]

$S_{7}$ = 49

=> $S_{7}$ = $\dfrac{7}{2}$ [2a + (7 - 1)d]

=> 49 = $\dfrac{7}{2}$ (2a + 6d)

=> 49 = $\dfrac{7}{2}$ × 2(a + 3d)

=> 49 = 7(a + 3d)

=> 7(a + 3d) = 49

=> a + 3d = 7

=> a = 7 - 3d ________ (eq 1)

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Similarly..

$S_{17}$ = 289

=> $S_{17}$ = $\dfrac{17}{2}$ [2a + (17 - 1)d]

=> 289 = $\dfrac{17}{2}$ (2a + 16d)

=> 289 = $\dfrac{17}{2}$ × 2 (a + 8d)

=> 289 = 17(a + 8d)

=> 17(a + 8d) = 289

=> a + 8d = 17

=> (7 - 3d) + 8d = 17 [From (eq 1)]

=> 5d = 10

=> d = 2

• Put value of d in (eq 1)

=> a = 7 - 3(2)

=> a = 7 - 6

=> a = 1

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• We have to find the sum of nth term.

So..

$S_{n}$ = $\dfrac{n}{2}$ [2a + (n - 1)d]

From above calculations we have d = 2 and a = 1

=> $S_{n}$ = $\dfrac{n}{2}$ [2(1) + (n - 1)2]

=> $S_{n}$ = $\dfrac{n}{2}$ (2 + 2n - 2)

=> $S_{n}$ = $\dfrac{n}{2}$ × 2n

=> $S_{n}$ = n × n

=> $S_{n}$ = n²

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Sum of first nth term is n².

____________ [ANSWER]

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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 that,

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

=> 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.