Question

Find the common difference of the A.P and write the next two terms: 75, 67, 59,51, ….

Answer 1

Hint: For solving this question you should know about the arithmetic progression. Here, as we know that in an arithmetic progression, terms are always in a fixed sequence. And that sequence will be calculated as we will subtract every ${{\left( n+1 \right)}^{th}}$ term from the ${{n}^{th}}$ term and in an A.P, it is always fixed for every term.

Complete step-by-step solution:
According to the question we have to find the common difference of the A.P and write the next two terms of this A.P. So, as we know that the A.P is written as $a+\left( n-1 \right)d$, where $a$ will be ${{a}_{1}}$ for the first term of this A.P and $n$ will be the number of terms which we are going to calculate and here, $d$ is the common difference between all the terms. We also know that the common difference is always the same for every term (next term) in an A.P. So, we can calculate any arithmetic progression by $a+nd$ if all the components are given or all the components are known. And we can find the term of any random number of any A.P by using this formula.
So, for calculation the common difference of this A.P,
$\begin{align}
  & \Rightarrow {{a}_{2}}-{{a}_{1}} \\
 & =67-75 \\
 & =-8 \\
\end{align}$
So, here $d=-8$.
And we have to find the next two terms of the A.P, which means we have to find the ${{5}^{th}}$ and ${{6}^{th}}$ terms. So, for the ${{5}^{th}}$ term,
$\begin{align}
  & {{a}_{5}}={{a}_{1}}+4\left( d \right) \\
 & \Rightarrow {{a}_{5}}=75+4\left( -8 \right)=75-32=43 \\
\end{align}$
Similarly we will find the ${{6}^{th}}$ term,
$\begin{align}
  & {{a}_{6}}={{a}_{1}}+5\left( d \right) \\
 & \Rightarrow {{a}_{5}}=75+5\left( -8 \right)=75-40=35 \\
\end{align}$
So, the required terms are 43 and 35.

Note: While solving the next terms of any A.P of any one term in an A.P, you have to always remember that the number of that term will be as $\left( n-1 \right)$ in the formula of A.P because the first term is already added in that and then if we again add this, then it will not be in A.P.