Question

a,2a-1,3a-2,4a-3,..... For a particular number a, the first term in the sequence above is equal to a and each term thereafter is 7 greater than the previous term. What is the value of the sixteenth term in the sequence?

Answer 1

Hint: If you observe the sequence $a,2a-1,3a-2,4a-3,.....$ , you will find that it is an AP with first term a and common difference (a-1). As it is given that each term is having a difference of 7, so equate (a-1) with 7 to get the value of a. Finally, use the formula ${{T}_{r}}=a+\left( r-1 \right)d$ to get the answer.

Complete step by step solution:
Before starting with the solution, let us discuss what an A.P. is. A.P. stands for arithmetic progression and is defined as a sequence of numbers for which the difference of two consecutive terms is constant. The general term of an arithmetic progression is denoted by ${{T}_{r}}$, and sum till r terms is denoted by ${{S}_{r}}$ .
${{T}_{r}}=a+\left( r-1 \right)d$
${{S}_{r}}=\dfrac{r}{2}\left( 2a+\left( r-1 \right)d \right)=\dfrac{r}{2}\left( a+l \right)$
Now moving to the situation given in the question. If you observe the sequence $a,2a-1,3a-2,4a-3,.....$ , you will find that it is an arithmetic progression with first term as a and common difference (a-1).
It is also given that the first term in the sequence above is equal to a and each term thereafter is 7 greater than the previous term, which means that the common difference is equal to (a-1).
$a-1=7$
$\Rightarrow a=8$
Therefore, the first term of the AP is 8 and the common difference of the AP is (a-1)=7.
Now, we will use the formula ${{T}_{r}}=a+\left( r-1 \right)d$ to find the sixteenth term of the AP.
${{T}_{16}}=a+\left( 16-1 \right)\left( a-1 \right)$
$\Rightarrow {{T}_{16}}=a+15\left( a-1 \right)$
Now we will put the value of a as found in the above results.
${{T}_{16}}=8+15\left( 8-1 \right)$
$\Rightarrow {{T}_{16}}=8+15\times 7=113$
Therefore, the answer to the above question is 113.

Note: Always, while dealing with an arithmetic progression, try to find the first term and the common difference of the sequence, as once you have these quantities, you can easily solve the questions related to the given arithmetic progressions. Also, remember that the answer must be reported in the simplest form, i.e., after substituting all the known values of the variable. For example: as we have reported the answer to the above question after putting the value of a.