Question

How do you write the rule of the nth term of the arithmetic sequence $a_{10} =2, a_{22}=-8.8$

Answer 1

Hint: The given question is about arithmetic progression, in which we need to find the equation for the nth term of the given series, here to solve this question we need to have the idea for the general equation for the nth term for any series given arithmetic progression.

Formula Used:
The nth term of the series can be given as:
\[ {a_n} = {a_1} + (n - 1)d\]
Here “d” is the common difference of the series

Complete step by step solution:
The given question needs to be solved by using the general equation of the nth term of the given arithmetic series, here we have to find the required quantities used in the equation by using the data given in the question.
The nth term of the series can be given as:
\[ {a_n} = {a_1} + (n - 1)d\]
Here “d” is the common difference of the series,
To find the common difference here we have to solve with the given two terms in the question,
Given data:
\[ \Rightarrow {a_{10}} = 2,\,{a_{22}} = - 8.8\]
\[ \Rightarrow {a_1} + 9d = 2,\,{a_1} + 21d = - 8.8\]
To solve the above equation here we subtract both the above expression, on solving we get:
\[
   \Rightarrow ({a_1} + 9d) - ({a_1} + 21d) = 2 - ( - 8.8) \\
   \Rightarrow 12d = - 10.8 \\
   \Rightarrow d = \dfrac{{ - 10.8}}{{12}} = - 0.9 \\
 \]
Substituting “d” in the above equation for getting the first term we get:
\[
   \Rightarrow {a_1} + 9d = 2 \\
   \Rightarrow {a_1} + 9( - 0.9) = 2 \\
   \Rightarrow {a_1} = 2 + 8.1 = 10.1 \\
 \]
General equation of nth term:
\[
   \Rightarrow {a_n} = {a_1} + (n - 1)d \\
   \Rightarrow {a_n} = 10.1 + (n - 1)( - 0.9) \\
   \Rightarrow {a_n} = 11 - 0.9n \\
 \]
Note: Here in the above question we have to find the equation for the nth term of the series given, for which we should know the equation for the series and then by simply finding the values which can be obtained by the given data in the question we can write the final general equation for the given series.