Question

Find the sum of the products of the corresponding terms of the sequences 2, 4, 8, 16, 32 and 128, 32, 8, 2, .

Answer 1

The sequence is obtained by multiplying the corresponding terms of the two sequences 2,4,8,16,32 and 128,32,8,2, 1 2

The obtained sequences after multiplying the corresponding terms are,

2×128,4×32,8×8,16×2,32× 1 2 256,128,64,32,16

The above sequence is in a G.P.

Let the first term and common ratio of the given G.P. be a and r respectively.

Here,

a=256 r= 128 256 = 1 2 n=5

Here, r<1.

The formula for the sum of first n terms of a G.P. for r<1 is given by,

S n = a( 1− r n ) 1−r (1)

Substitute the values of a and r in equation (3) to obtain the sum of first five terms.

S 5 = 256( 1− ( 1 2 ) 5 ) 1−( 1 2 ) = 256( 1−( 1 32 ) ) 1 2 =256×2×( 32−1 32 ) =16×31

Further simplify the above expression.

S 5 =16×31 =496

Thus, the sum of the sequence obtained by multiplying the corresponding terms of the two sequences 2,4,8,16,32 and 128,32,8,2, 1 2 is 496.