Question

Which term of the sequence $ 4 $ , $ 9 $ , $ 14 $ , $ 19 $ , …… is $ 124 $ ?
(A) $ 24 $
(B) $ 25 $
(C) $ 26 $
(D) $ 27 $

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 number of terms that correspond to a given term in an arithmetic progression, we must know the formula for a general term in an AP: $ {a_n} = a + \left( {n - 1} \right)d $ .

Complete step-by-step answer:
So, we have, $ 4 $ , $ 9 $ , $ 14 $ , $ 19 $ , ……
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.
Now, we are given a term in the AP as $ 124 $ and we have to find the number of terms in the AP that corresponds to $ 124 $ .
Here, first term $ = a = 4 $ .
Now, we can find the common difference of the arithmetic progression by subtracting any two consecutive terms in the series.
So, common ratio \[ = d = 9 - 4 = 5\]
So, $ d = 5 $ .
We know the formula for the general term in an arithmetic progression is $ {a_n} = a + \left( {n - 1} \right)d $ .
So, considering $ 124 $ as the nth term of the AP, we can find the number of terms in the AP that corresponds to $ 124 $ .
So, \[{a_n} = a + \left( {n - 1} \right)d = 124\]
Substituting the values of a and d in the above equation, we get,
 $ \Rightarrow 4 + \left( {n - 1} \right)5 = 124 $
Now, in order to solve the above equation for the value of n, we shift all the constant terms to the right side of the equation. So, we get,
 $ \Rightarrow \left( {n - 1} \right)5 = 124 - 4 $
Carrying out the calculations and dividing both sides of the equation by $ 5 $ , we get,
 $ \Rightarrow \left( {n - 1} \right) = \dfrac{{120}}{5} $
 $ \Rightarrow \left( {n - 1} \right) = 24 $
Adding $ 1 $ to both sides of the equation, we get,
 $ \Rightarrow n = 25 $
So, $ 124 $ is the $ {25^{th}} $ term of the sequence $ 4 $ , $ 9 $ , $ 14 $ , $ 19 $ , …… .
Hence, option (B) is correct.
So, the correct answer is “Option B”.

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 $ .