Answer:

n2

Explanation:

Sum of the first n terms of an AP is given by Sₙ = n/2 [2a + (n - 1) d] or Sₙ = n/2 [a + l], and the nth term of an AP is aₙ = a + (n - 1)d

Here, a is the first term, d is the common difference and n is the number of terms and l is the last term.

Given,

Sum of the first 7 terms, S₇ = 49

Sum of the first 17 terms, S₁₇ = 289

We know that sum of n terms of AP is Sₙ = n/2 [2a + (n - 1) d]

S₇ = 7/2 [2a + (7 - 1)d]

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

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

S₁₇ = 17/2 [2a + (17 - 1) d]

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

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

Subtracting equation (i) from equation (ii),

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

5d = 10

d = 2

From equation (i),

7 = a + 3 × 2

7 = a + 6

a = 1

Sₙ = n/2 [2a + (n - 1) d]

= n/2 [2 × 1 + (n - 1) 2]

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

= n/2 × 2n

= n2

Answer:

search-icon-header

Search for questions & chapters

search-icon-image

Question

Bookmark

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.

Medium

Solution

verified

Verified by Toppr

Given

S

7

=49 and S

17

=289

By using S

n

=

2

n

[2a+(n−1)d] we have,

S

7

=

2

7

[2a+(7−1)d]=49

⇒49=

2

7

[2a+(7−1)d]

⇒49=

2

7

(2a+6d)

⇒7=a+3d

⇒a+3d=7...................(i)

S

17

=

2

17

[2a+(17−1)d]=289

⇒289=

2

17

[2a+(17−1)d]

⇒289=

2

17

(2a+16d)

⇒17=a+8d

⇒a+8d=17......................(ii)

Substituting (i) from (ii), we get

5d=10 or d=2

From equation (i),

a+3(2)=7

a+6=7 or a=1

S

n

=

2

n

[2(1)+(n−1)2]

=

2

n

[2+(n−1)2]

=

2

n

(2+2n−2)=n

2

hey i m paresh