Question

How do you find \[{a_{43}}\] given 5,9,13,17……?

Answer 1

Hint: Arithmetic Progression (AP) is a sequence of numbers in order in which the difference of any two consecutive numbers is a constant value. The finite portion of an AP is known as finite AP and therefore the sum of finite AP is known as arithmetic series. In the given question, we are asked to find the forty-third term of an Arithmetic sequence, hence we need to apply the formula of nth term of an AP i.e., \[{a_n} = {a_1} + \left( {n - 1} \right) \times d\] , in which we need to substitute all values in the formula from the given sequence, to get the value of \[{a_{43}}\] .

Formula used:
 \[{a_n} = {a_1} + \left( {n - 1} \right) \times d\]
 \[{a_n}\] is the nth term in the sequence
 \[{a_1}\] is the first term in the sequence
 \[n\] is number of terms
 \[d\] is the common difference between terms
 \[d = {a_2} - {a_1}\]
 \[{a_2}\] is the second term in the sequence
 \[{a_1}\] is the first term in the sequence

Complete step by step solution:
Given,
5,9,13,17…...
In which we need to find \[{a_{43}}\] .
From the given terms, the first term of the sequence is \[{a_1} = 5\] and the common difference is:
 \[d = {a_2} - {a_1}\]
 \[d = 9 - 5 = 4\] ,
Hence, to calculate the forty-third term i.e., n=43; we have the formula as:
 \[{a_n} = {a_1} + \left( {n - 1} \right) \times d\]
If we substitute the given data we have:
 \[ \Rightarrow {a_{43}} = 5 + \left( {43 - 1} \right) \times 4\]
Evaluating the terms, we get:
 \[ \Rightarrow {a_{43}} = 5 + \left( {42} \right) \times 4\]
 \[ \Rightarrow {a_{43}} = 5 + 168\]
 \[ \Rightarrow {a_{43}} = 173\]
Therefore, the value of the 43rd term is \[{a_{43}} = 173\] .
So, the correct answer is “ \[{a_{43}} = 173\] ”.

Note: If we are asked to find the Sum of N Terms of arithmetic progression, then for an AP, the sum of the first n terms is calculated if the first term and the total terms are known. The formula for the arithmetic progression sum for n terms is:
 \[S = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right] \]