Question

In series \[357,\text{ }363,\text{ }369,\ldots .\] What will be the \[{{10}^{th}}\] term?
A. $ 405 $
B. $ 411 $
C. $ 413 $
D. $ 417 $

Answer 1

Hint: We know that in order to find the \[{{10}^{th}}\] term of the given infinite series we need to first identify the series and then use the formula for its nth term to find the \[{{10}^{th}}\] term. The nth term is also known as the general term of the pattern.

Complete step-by-step answer:
A pattern is a series of sequences that repeats. There are two main types of math patterns, number patterns or sequence of numbers arranged according to a rule or rules and shape patterns, which are labelled by using letters and the way that they repeat. It helps to calculate any particular term of the pattern without writing the actual pattern. By putting the given value of in the nth term formula of the given pattern to get the asked or respective term of the pattern. An arithmetic progression is given as:
 $ a,\left( a+d \right),\left( a+2d \right),\left( a+3d \right).........a+\left( n-1 \right)d $ with a as the first term and d as the common difference. The nth term of this series will be given as: $ {{a}_{n}}=a+(n-1)d. $
\[363-357=6;369-363=6............\]
That is the difference between the next term and the previous term is the same for all the consecutive terms and the first term is 4. Therefore, we can conclude that the given series is an Arithmetic Series Now by using Arithmetic Progression the given series is an Arithmetic Progression in which \[a=357\]and \[d=6.\] Thus, $ {{10}^{th}} $ term is given by
 $ \left[ a+\left( 10-1 \right)d \right]=\left[ a+9d \right] $
On substitution we get;
 $ \Rightarrow 357 + ( 9\times6) = 357 + 54 = 411 $
So, the correct answer is “Option B”.

Note: Remember that by identifying the series correctly is essential, since if the series is not identified correctly then the whole calculation that follows will be incorrect. The sum of n terms of an Arithmetic Progression is given by $ {{S}_{n}}. $