Question

Write the sequence with ${{n}^{th}}$ term ${{a}_{n}}=9-5n$ . Show that all the sequences form an A.P.

Answer 1

- Hint: For solving this question first we will understand what we mean by arithmetic progression and then we will prove that the sequence given in the problem will form an arithmetic progression.

Complete step-by-step solution -
Given:
It is given that ${{n}^{th}}$ term of a sequence is ${{a}_{n}}=9-5n$ and we have to prove that this sequence will be in arithmetic progression.
Now, first, we will understand when a sequence is called an A.P. and what are important conditions for a sequence to be in arithmetic progression.
Arithmetic Progression:
In a sequence when the difference between any two consecutive terms is equal throughout the series then, such sequence will be called to be in arithmetic progression and the difference between consecutive terms is called as the common difference of the arithmetic progression. If ${{a}_{1}}$ is the first term of an A.P. and common difference of the A.P. is $d$ then, ${{n}^{th}}$ term of the A.P. can be written as ${{a}_{n}}={{a}_{1}}+d\left( n-1 \right)$ .
Now, from the above expression of ${{n}^{th}}$ term of an A.P. we can conclude that for any sequence if the expression of ${{n}^{th}}$ term is a linear expression of $n$ then, that sequence will be an A.P. and the coefficient of $n$ in the expression of ${{n}^{th}}$ term will be the common difference of the arithmetic progression.
Now, we come back to our question in which ${{n}^{th}}$ term of a sequence is ${{a}_{n}}=9-5n$ . And, from the above discussion we can say that the given sequence will be in A.P. as expression of ${{n}^{th}}$ term of the sequence is a linear expression of $n$ and the common difference of the A.P. will be $-5$ .
Given sequence is written below:
$\begin{align}
  & {{a}_{n}}=9-5n \\
 & \Rightarrow {{a}_{1}}=9-5=4 \\
 & \Rightarrow {{a}_{2}}=9-10=-1 \\
 & \Rightarrow {{a}_{3}}=9-15=-6 \\
\end{align}$
Thus, the given sequence will be $\left\{ 4,-1,-6,-11,-16,-21,................,\left( 9-5n \right) \right\}$ and it will be in A.P. whose common difference is $-5$ and the first term will be $4$ .
Hence, proved.

Note: Here, the student should know the concept of A.P. and how to express the general expression of ${{n}^{th}}$ term of an A.P. and important point should be remembered so, that question can be answered quickly and correctly without any confusion.