Question

If we have $n^{th}$ term as ${{a}_{n}}=5-11n$ , then the common difference will be___________

Answer 1

Hint: Here we have been given an A.P whose ${{n}^{th}}$ term is given we have to find the common difference. Firstly we will find the first and the second value of the A.P by using the ${{n}^{th}}$ value formula given by substituting different values of $n$. Then we will subtract the two values obtained and get the common difference which is our desired answer.

Complete step-by-step solution:
The ${{n}^{th}}$ term is given as follows:
${{a}_{n}}=5-11n$
Firstly we will find the first term of the A.P by putting $n=1$ in above values,
${{a}_{1}}=5-11\times 1$
$\Rightarrow {{a}_{1}}=5-11$
So we get the value as,
${{a}_{1}}=-6$
Next we will put $n=2$ in the value given,
${{a}_{1}}=5-11\times 2$
$\Rightarrow {{a}_{1}}=5-22$
So we get the value as,
${{a}_{1}}=-17$
So we got the first two term of the A.P as,
$-6,-17,....$
As we can see that it is an increasing A.P so the common difference is the subtraction between the first and second value as follows:
Common difference $d=-17-\left( -6 \right)$
$\Rightarrow d=-17+6$
$\Rightarrow d=-11$
Hence the common difference will be $-11$.

Note: The given value represents an A.P as we will find the terms we will see the same pattern among the consecutive terms that is the difference between the consecutive terms is the same. Even if we can’t understand what is the name of the value given, we can simply put a different value for the unknown variable and see what pattern is there. There is ${{n}^{th}}$ term, sum of $n$ terms formula for an A.P which makes it easy to get solutions for complicated questions. If we take a portion of finite arithmetic progression we call it finite arithmetic progression and the sum of it is called the arithmetic series.