Question

What is the \[20^{th}\] term of the A.P \[13,26,39,...\]?
(A) \[130\]
(B) \[260\]
(C) \[390\]
(D) \[420\]

Answer 1

Hint: : In the above question, we have to find out the twentieth term of an AP. To find this we should know the first term of an AP and the common difference between two consecutive terms of an AP. Also, if you want to form an AP then add the fixed number to any of the terms of AP and a series will be obtained.

Complete step by step answer:
AP stands for arithmetic progression. Arithmetic progression in mathematics is the sequence of numbers and if we find the difference between any two consecutive values of AP then their difference gives us the constant value.
The behavior of arithmetic progressions is decided by the common difference between two consecutive terms of an AP. An arithmetic progression can be both finite and infinite. In arithmetic progression, the common difference can be of two types, positive common difference, and negative common difference. If the common difference is positive then the terms will move towards positive infinity but if the common difference is negative then the terms will move towards negative infinity.
Geometric progression is also one term in mathematics and it is the sequence in which the ratio between two consecutive terms always remain constant. The formula for geometric progression is given as shown below.
\[{{a}_{n}}=a{{r}^{n-1}}\]
Where ‘r’ is the common ratio between two consecutive terms.
In the question, we have to find the \[20^{th}\] term in the AP \[13,26,39,...\]
As we know that the \[{{n}^{th}}\] term of an AP is given as shown below.
\[{{t}_{n}}=a+(n-1)d\]
Where \[{{t}_{n}}\] represents the \[{{n}^{th}}\] term of the AP, ‘a’ represents the first term of an AP, and ‘d’ is the common difference between two consecutive terms which remains constant.
So in the above question, the first term of the AP is \[13\].
\[a=13\]
And ‘d’ is the common difference between two consecutive terms and n is \[20\].
\[\begin{align}
  & d=26-13 \\
 & \Rightarrow d=13 \\
\end{align}\]
And we have to find the \[{{20}^{th}}\] term which is as follows.
\[\begin{align}
  & {{t}_{20}}=13+(20-1)13 \\
 & \Rightarrow {{t}_{20}}=13+19\times 13 \\
 & \Rightarrow {{t}_{20}}=13+247 \\
 & \Rightarrow {{t}_{20}}=260 \\
\end{align}\]
Hence the \[{{20}^{th}}\] term of the AP will be \[260\].

So, the correct answer is “Option B”.

Note:
In mathematics, there are three types of progression, the first one is arithmetic progression, the second one is a geometric progression, and the third one is harmonic progression. If we take the reciprocals of an arithmetic progression, then harmonic progression will be obtained.