Question

The number of terms common between the series $1,2,4,8,...{\rm{to\, }}100{\rm{\,terms}}$ and $1,4,7,10,...{\rm{to\, }}100{\rm{\, terms}}$ is
A.6
B.4
C.5
D.None of these

Answer 1

Here in this question firstly we will find the general form of the series in terms of variable for both the series. Then we will see the pattern of which terms will be repeating in both the series. Then we will get the maximum value of both the series. Then we will see for how many terms of that pattern are there in both the series.

Complete step-by-step answer:
Given the 1st series of numbers is $1,2,4,8,...{\rm{to \,}}100{\rm{\,terms}}$ and the 2nd series of numbers is $1,4,7,10,...{\rm{to\, }}100{\rm{\, terms}}$
Firstly we will write the general form of the series in terms of variable for both the series.
So, the general form of the series in terms of variable for the 1st series is${{\rm{2}}^{n - 1}}$. Similarly the general form of the series in terms of variables for the 2nd series is $1 + \left( {m - 1} \right)3 = 3m - 2$.
From this general form we can clearly see that when a term of the 2nd series is divided by 3 then remainder will be 1 for every term in the 2nd series. So, now we have to find the terms in the first terms which on dividing by 3 gives the remainder of 1. Therefore
When ${2^0}$ is divided by 3 will give 1 as the remainder.
When ${2^1}$ is divided by 3 will give 2 as the remainder.
When ${2^2}$is divided by 3 it will give 1 as the remainder.
When ${2^3}$ is divided by 3 will give 2 as the remainder.
And then this pattern will follow as all the even power in this series will give a remainder of 1 and all the odd power in this series will give a remainder of 2.
Now we will calculate the maximum value of the 2nd series. So we get
100th term of the 2nd series$= 3 \times 100 - 2 = 298$
So we can see that for even the 10th term of the first series is ${{\rm{2}}^{10 - 1}} = {2^9} = 512$ which is greater than the last term of the 2nd series. So, only for the first 9 terms the common numbers are present.
Therefore, for all the even power of the series is the common terms on both the series. So ${2^0},{2^2},{2^4},{2^6},{2^8}$ are the only terms with the even power for the first 9 terms of the series.
Hence, these 5 terms are the terms which are common in both the series.
So, option C is the correct option.

Note: Here we have to note that the general form of the series in terms of variable is the form or the equation where by putting the values of the variable as integers we will get the original series. we have to make sure that while finding the terms which when divided will give 1 as the remainder must be divided by a same number i.e. 3 in case of this question. Also we have to note that the 1st series is increasing exponentially that is why out of the 100 terms there were the common terms only in the first 9 numbers of both the series.