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 .
hey frnds plzz help me ...

Answer 1

♠ ANSWER :

Given,

S₇ = 49

S₁₇ = 289

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

S₇ = 7 / 2 [ 2a + ( 7 - 1 ) d ]

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

14 = 2 ( a + 3d )

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

S₁₇ = 17 / 2 [ 2a + ( 17 - 1 ) d ]

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

17 * 2 = 2 ( a + 8 d )

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

Subtracting equation ( i ) from ( ii ),

( a + 8d ) - ( a + 3d ) = 17 - 7

a + 8d - a - 3d = 10.

5d = 10

d = 10 / 5 = 2.

⏺️So, Common difference is 2.

Putting value of d in equation ( i ),

7 = a + 3d

7 = a + 3 *2

a = 7 - 6 =1.

⏺️Now, Sum of first n terms of A. P.

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

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

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

Sn = n / 2 [ 2 n]

Sn = n *n

➤ Sn = n²

Answer 2

$s7 = \frac{7}{2} (2a + 6d)$

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

$\frac{49 \times 2}{7} = 2a + 6d$

$14 = 2a + 6d$

$2(a + 3d) = 14$

$a + 3d = 7$




$s17 = \frac{17}{2} (2a + 16d)$

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

$\frac{289 \times 2}{17} = 2a + 16d$

$34 = 2a + 16d$

$34 = 2(a + 8d)$

$a + 8d = 17$




$(a+3d)-(a+8d) = 7-17$

$a-a-8d+3d=-10$

$- 5d = - 10$

$d= 2$


$a + 3d = 7$

$a + 6 = 7$

$a = 1$



$sn = \frac{n}{2} (2 \times 1 + (n - 1) \times 2)$

$sn = \frac{n}{2} (2 + 2n - 2)$

$sn = \frac{n}{2} \times 2(1 - 1 + n)$

$sn = n \times n$
$sn = {n}^{2}$