Question

The number of terms common between the sum $1 + 2 + 4 + 8 + .....$to $100$ terms and $1 + 4 + 7 + 10 + .....$to $100$ terms is

• A. 6
• B. 4
• C. 5
• D. none of these.

Answer 1

Answer: (C)

5

Answer 2

For a $G.P. 1+2+4+8+.....T_{n} = 2^{n-1}$ For an $A.P. 1+4+7+10+.....$ $T_{m} = 1+\left(m-1\right)3 =3m-2$ They are common if $2^{n-1} = 3m-2\, or \,2^{n-1} +2 = 3m$ i.e., $2^{n-2} +1 = \frac{3m}{2}\le\frac{3}{2}\left(100\right) = 150$ $\Rightarrow n\le9, m\le100$ By trial, $n=1, m=1 ; n=3, m=2 ; n=5,$ $m=6, n=7, m=22; n=9,m=86$