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

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$

The number of terms common between the sum $1 + 2

Notes on Arithmetic Progression

Concepts Used:

Arithmetic Progression

Arithmetic Progression (AP) is a mathematical series in which the difference between any two subsequent numbers is a fixed value.

For example, the natural number sequence 1, 2, 3, 4, 5, 6,... is an AP because the difference between two consecutive terms (say 1 and 2) is equal to one (2 -1). Even when dealing with odd and even numbers, the common difference between two consecutive words will be equal to 2.

In simpler words, an arithmetic progression is a collection of integers where each term is resulted by adding a fixed number to the preceding term apart from the first term.

For eg:- 4,6,8,10,12,14,16

We can notice Arithmetic Progression in our day-to-day lives too, for eg:- the number of days in a week, stacking chairs, etc.

Read More: Sum of First N Terms of an AP

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

The Correct Option is C

Solution and Explanation

Top Questions on Arithmetic Progression

Notes on Arithmetic Progression

Concepts Used:

Arithmetic Progression