Question

If mth terms of the series $63 + 65 + 67 + 69 + ....$ and $3 + 10 + 17 + 24 + ...$ be equal, then m equals
(A) $11$
(B) $12$
(C) $13$
(D) $15$

Answer 1

Hint: The given problem involves the concepts of arithmetic progression. The first few terms of the series are given to us in the problem. We can find out the common difference of an arithmetic progression by knowing the difference of any two consecutive terms of the series. For finding out the nth term of an arithmetic progression, we must know the formula for the general term in an AP: ${a_n} = a + \left( {n - 1} \right)d$.

Complete step-by-step solution:
So, we have the first series as $63 + 65 + 67 + 69 + ....$
We can notice that the difference of any two consecutive terms of the given series is constant. So, the given sequence is an arithmetic progression.
Here, first term $ = a = 63$.
Now, we can find the common difference of the arithmetic progression by subtracting any two consecutive terms in the series.
So, common ratio \[ = d = 65 - 63 = 2\]
So, $d = 2$ .
We know the formula for the general term in an arithmetic progression is ${a_n} = a + \left( {n - 1} \right)d$.
So, mth term of this AP will be ${a_n} = a + \left( {m - 1} \right)d$
Substituting the values of a and d in the expression, we get,
\[ \Rightarrow {a_m} = 63 + \left( {m - 1} \right) \times 2\]
$ \Rightarrow {a_m} = 63 + 2m - 2$
$ \Rightarrow {a_m} = 2m + 61$
Now, we have the first series as $3 + 10 + 17 + 24 + ...$
We can notice that the difference of any two consecutive terms of the given series is constant. So, the given sequence is an arithmetic progression.
Here, first term $ = a = 3$.
Now, we can find the common difference of the arithmetic progression by subtracting any two consecutive terms in the series.
So, common ratio \[ = d = 10 - 3 = 7\]
So, $d = 7$ .
So, mth term of this AP will be \[{a_m} = a + \left( {m - 1} \right)d\]
Substituting the values of a and d in the expression, we get,
$ \Rightarrow {a_m} = 3 + \left( {m - 1} \right) \times 7$
$ \Rightarrow {a_m} = 3 + 7m - 7$
$ \Rightarrow {a_n} = 7m - 4$
Now, we are given that mth terms of both the arithmetic progressions are equal. So, we get,
Equating both the expressions, we get,
$7m - 4 = 2m + 61$
Taking the variables to the left side of equation and constant terms to right side of the equation, we get,
$ \Rightarrow 7m - 2m = 61 + 4$
$ \Rightarrow 5m = 65$
Dividing both sides by $5$, we get,
$ \Rightarrow m = 13$
Hence, the value of m is $13$.
Therefore, the option (C) is the correct answer.

Note: Arithmetic progression is a series where any two consecutive terms have the same difference between them. The common difference of an arithmetic series can be calculated by subtraction of any two consecutive terms of the series. Any term of an arithmetic progression can be calculated if we know the first term and the common difference of the arithmetic series as: ${a_n} = a + \left( {n - 1} \right)d$. We must take care of the calculations so as to be sure of the final answer. We also should know the transposition rule to find the value of x by shifting terms in the equation.