Question

The 17th term of an AP exceeds its 10th term by 7. Find the common difference.

Answer 1

Here in this question, we have to find the common difference of Arithmetic Progression sequence. To find this mainly by using the formula ${a_n} = a + \left( {n - 1} \right)d$ where $a$ is the first term, ${a_n}$ is the nth term and d is the common difference of an Arithmetic Progression. Using the formula, we have to find the 17 and 10 terms by given value as 17 and 10 in the formula later by the condition that the 17th term of an AP exceeds its 10th term by 7, on simplification, we get the required value.

Complete step by step answer:
An arithmetic progression (A.P.), also called an arithmetic sequence, is a sequence of numbers which differ from each other by a common difference. For example, the sequence 2, 4, 6, 8,… is an arithmetic sequence with the common difference 2.
We can find the common difference of an AP by finding the difference between any two adjacent terms. And
Formula to find the nth term of an A.P. is
${a_n} = a + \left( {n - 1} \right)d$
where ${a_n}$= nth term, a= the first term and d is a common difference.
Now we have to find the 17th term of an A.P. by giving n=17, then
$\Rightarrow \,\,{a_{17}} = a + \left( {17 - 1} \right)d$
$\Rightarrow \,\,{a_{17}} = a + 16d$-(1)
And find the 10th term of an A.P. by giving n=10, then
$\Rightarrow \,\,{a_{10}} = a + \left( {10 - 1} \right)d$
$\Rightarrow \,\,{a_{10}} = a + 9d$-(2)
Given the condition
17th term of an A.P. exceeds its 10th term by 7. i.e.,
$\Rightarrow \,\,{a_{17}} - {a_{10}} = 7$
On substituting equation (1) and (2), we get
$\Rightarrow \,\,\left( {a + 16d} \right) - \left( {a + 9d} \right) = 7$
$\Rightarrow \,\,a + 16d - a - 9d = 7$
On simplification, we get
$\Rightarrow \,\,7d = 7$
Divide both side by 7, then
$\Rightarrow \,\,d = 1$

Hence, the common difference is d=1.

Note: By considering the formula of arithmetic sequence we verify the common difference which we obtained. We have to check the common difference for all the terms. Suppose if we check for the first two terms not for other terms then we may go wrong. So definition of arithmetic sequence is important to solve these kinds of problems.