If the sum of first 7terms of an AP is 49 and that of 17 terms is 289 , find the sum of first n terms
Answers
Step-by-step explanation:
Given :-
The sum of first 7terms of an AP is 49 and that of 17 terms is 289 .
To find :-
Find the sum of first n terms ?
Solution :-
We know that
Sum of first n terms of an AP is
Sn = (n/2)[2a+(n-1)d]
Where, a = First term
d = Common difference
n = number of terms
Given that
The sum of first 7 terms = S 7 = 49
=> S 7 = (7/2)[2a+(7-1)d] = 49
=> (7/2)(2a+6d) = 49
=> (7/2)×2(a+3d) = 49
=> 7(a+3d) = 49
=> a+3d = 49/7
=> a+3d = 7 -------------------(1)
and
The sum of first 7 terms = S 17 = 289
=> S 17 = (17/2)[2a+(17-1)d] = 289
=> (17/2)(2a+16d) = 289
=> (17/2)×2(a+8d) = 289
=> 17(a+8d) = 289
=> a+8d = 289/17
=> a+8d = 17 -------------------(2)
On subtracting (1) from (2) then
a+8d = 17
a+3d = 7
(-) (-) (-)
________
0+5d = 10
________
=> 5d = 10
=> d = 10/5
=> d = 2
Common difference = 2
On Substituting the value of d in (1)
=> a+3(2) = 7
=> a +6 = 7
=>a = 7-6
=> a = 1
First term = 1
Now,
The sum of first n terms =(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 = (2n×n)/2
=> Sn = n×n
=> Sn = n²
Therefore, Sn = n²
Shortcut:-
Given that
S 7 = 49
=> S 7 = 7²
and
S 17 = 289
=> S 17 = 17²
Now,
Sn = n²
Answer:-
The sum of the first n terms of the given AP is n²
Used formulae:-
→ Sum of first n terms of an AP is
Sn = (n/2)[2a+(n-1)d]
Where, a = First term
d = Common difference
n = number of terms