Question

Find if 1000 is a term of the A.P. 25, 28, 31…….or not?

Answer 1

Hint: First of all find the first term and common difference of the given A.P. First term is the first term of the series and the common difference is calculated by taking any term and doing the subtraction of this term from its successor term. Now, we know that the general term of the A.P. is equal to $ {{T}_{n}}=a+\left( n-1 \right)d $ . In this general term “a” is the first term and “d” is the common difference and n is the nth term. Now, substitute 1000 in place of $ {{T}_{n}} $ and put the values of “a” and “d” that we have calculated above in this general formula and see whether the value of n is a positive integer or not.

Complete step-by-step answer:
The arithmetic progression given in the above question is:
25, 28, 31…….
The first term of the above A.P. is 25 which we are denoting as “a”.
The common difference of the above A.P. is calculated by taking any term and then subtracts this term from its successor term so let us take 25. Now, the successor term from 25 is 28 so subtracting 25 from we get,
 $ 28-25=3 $
From the above, the common difference of A.P. is 3. We are denoting a common difference as “d”.
We know that the general term for an A.P. is given by:
 $ {{T}_{n}}=a+\left( n-1 \right)d $
In the above formula, “a” is the first term, “d” is the common difference and “n” is the nth term of an A.P.
Now, we have to show whether 1000 belongs to the given A.P. or not so we are substituting 1000 in place of $ {{T}_{n}} $ in the general term of an A.P. and substituting “a” as 25 and “d” as 3 we get,
 $ \begin{align}
  & 1000=25+\left( n-1 \right)3 \\
 & \Rightarrow 1000=25+3n-3 \\
 & \Rightarrow 1000=22+3n \\
\end{align} $
Subtracting 22 on both the sides we get,
 $ 978=3n $
Dividing 3 on both the sides we get,
 $ \begin{align}
  & \dfrac{978}{3}=n \\
 & \Rightarrow 326=n \\
\end{align} $
As you can see that the value of n that we are getting is a positive integer which shows that 1000 belongs to the given A.P.

Note: The point to be taken care of is that while calculating the common difference of an A.P. the difference of the two terms is done in such a way that subtract the selected term from the successor term not subtraction of the selected term from the predecessor term. If you make such a mistake then the whole solution will be changed. In the below, we illustrate this problem by taking above A.P.
25, 28, 31…….
Let us take 28 so to find the common difference subtract 28 from 31 not 28 from 25. If we subtract 28 from 31 then we get 3 and when we subtract 28 from 25 then we get -3 so now you can see the big change that could happen if you make such a mistake.