Question

Find the next term of the series: - 20, 29, 38, 47, ?
(a) 56
(b) 52
(c) 59
(d) 58
(e) none of these

Answer 1

Hint: First of all check whether this is an AP or not. After that assume the first term of the given series as ‘a’ and the common difference as ‘d’. Find the value ‘d’ by subtracting \[{{1}^{st}}\] term from the \[{{2}^{nd}}\] term. To find the fifth term, apply the formula of A. P or arithmetic progression given as: - \[{{T}_{5}}=a+4d\], where \[{{T}_{5}}={{5}^{th}}\]term.

Complete step by step answer:
Here, we have been given the series 20, 29, 38, 47, …… and we have been asked to find the \[{{5}^{th}}\] term or the term after 47. Let us check if the given series is A. P or not.
We know that in A. P we have a common difference among successive terms.
\[\Rightarrow {{T}_{2}}-{{T}_{1}}={{T}_{3}}-{{T}_{2}}={{T}_{4}}-{{T}_{3}}=.....\]
Here, \[{{T}_{n}}={{n}^{th}}\]term
Now, \[{{T}_{2}}-{{T}_{1}}=29-20=9\]
\[{{T}_{3}}-{{T}_{2}}=38-29=9\]
\[{{T}_{4}}-{{T}_{3}}=47-38=9\]
Clearly, we can see that 9 is the common difference. Therefore, the given series is clearly an arithmetic progression.
Let us assume the first term of the series as ‘a’ and the common difference as ‘d’. We know that \[{{n}^{th}}\] term of the A. P is given by the relation: -
\[\Rightarrow {{T}_{n}}=a+\left( n-1 \right)d\]
Substituting a = 20 and d = 9 in the above relation, we get,
\[\begin{align}
  & \Rightarrow {{T}_{5}}=a+\left( 5-1 \right)d \\
 & \Rightarrow {{T}_{5}}=20+4\times 9 \\
 & \Rightarrow {{T}_{5}}=20+36 \\
 & \Rightarrow {{T}_{5}}=56 \\
\end{align}\]
Therefore, the term after 47 or the fifth term is 56.

So, the correct answer is “Option A”.

Note: Here, as you can see that every term is succeeding its previous term by 9. So, we can directly get the answer by adding 9 with 47 to get the answer 56. Actually, this direct answer which we will get is also the concept of arithmetic progression. So, we applied the formula of A. P to get the answer.