Question

if the 8th term of an A.P. is 37 and the 15th term is 15 more than the 12th term , find the A.P. . hence, find the sun of first 15 terms of the A.P.

Answer 1

Given that 8th term of an A.P. is 37 and the 15th term is 15 more than the 12th term 

We know that 8th term will be a+ 7d 
Thus, a + 7d = 37 ..... (1)

15term = a + 14d and 12th term= a + 11d

ATQ,  

a+ 14 d = a + 11d + 15
3d= 15
d= 5 

Putting d=5 in eq. 1, we get

a + 35 = 37
a= 2

So the sum of first 15 terms will be

S15= 15/2 ( 2* 2 + 14 * 5)
= 15/2 ( 4 + 70)
= 15/2 * 74
=15 * 37
=555

Answer 2

Let a = first term , d = common difference

eighth term = a8 = a + 7d = 37    ........(1)

Also 15th​ term = 15 + 12th term

⇒a15 = 15 + a12⇒a + 14d = 15 + a + 11d⇒14d − 11d = 15⇒3d = 15⇒d = 5from (1), we get a + 7×5 = 37so, a + 35 = 37a = 2.so the required AP is a, a + d, a + 2d,......... or 2, 7, 12, ..........We know that, Sn = n2[2a + (n−1)d]S15 = 152[2(2) + 14×5]S15 = 15(2 + 35) = 15 × 37 = 555