Question

In an A.P, the first term is 2 and the sum of the first five terms is one-fourth of the next five terms. Show that 20 th term is –112.

Answer 1

The first term of A.P. is 2. Also, the sum of first five terms is one-fourth of the next five terms.

Let a, d be the first term and common difference of the given A.P.

a=2

As the above sequence is in A.P., the terms are given by,

2, 2+d, 2+2d, 2+3d, 2+4d......

Let S 1 , S 2 be the sum of first five terms and next five terms of an A.P.

Sum of the first five terms is,

S 1 =2+2+d+2+2d+2+3d+2+4d =10+10d (1)

Similarly the sum of next five terms is given by,

S 2 =2+5d+2+6d+2+7d+2+8d+2+9d =10+35d (2)

According to the given question,

S 1 = 1 4 × S 2

Substitute the values of S 1 and S 2 from equation (1) and equation (2) respectively in the above expression.

10+10d= 1 4 ×( 10+35d ) 4×( 10+10d )=10+35d 40+40d=10+35d 5d=−30

Further simplify the above expression.

d= −30 5 =−6

The formula to find terms in an A.P. is given by,

T n =a+( n−1 )d

Substitute the values of a, dand n as 2, −6, 20 in the above expression.

T 20 =2+( 20−1 )×5 =2+19×( −6 ) =2−114 =−112

Thus, the 20 th term of the A.P. is −112.