Question

The first term of an arithmetic progression is -7 and the common difference is 5. Find its 18th term and the general term.

Answer 1

Hint: The formula for writing \[{{n}^{th}}\] term of an arithmetic progression is
\[{{n}^{th}}\ term=a+(n-1)d\] (Where ‘a’ is the first term and‘d’ is the common difference of the arithmetic progression).
For the calculation of the 18th term, we just need to put the values of ‘a’ and ‘d‘in the formula that is above and then put n=18.
For the calculation of the general term, we just need to put the values of ‘a’ that is the first term and ‘d’ that is the common difference of the arithmetic progression in the formula of the term that is given above.

Complete step-by-step answer:
As mentioned in the question, it is given that the first term of the arithmetic progression is -7, therefore ‘a’ is equal to -7. It is also given in the question that the common difference of the arithmetic progression is 5, therefore the value of ‘d’ is 5.
Now, using the formula for the \[{{n}^{th}}\] term of an arithmetic progression, we can write as following
 \[{{n}^{th}}\ term=a+(n-1)d\]
 Now, for this particular arithmetic progression , the general term can be written as
\[\begin{align}
  & =-7+(n-1)5 \\
 & =-7+5n-5 \\
 & =5n-12\ \ \ \ \ ...(a) \\
\end{align}\]
Hence, this is the general term for the arithmetic progression.
Now, for finding the 18th term of the given arithmetic progression, we can use the general term that is given in the equation (a) by substituting n=18 we get,
\[\begin{align}
  & =5n-12 \\
 & =5\times 18-12 \\
 & =90-12 \\
 & =78\ \ \\
\end{align}\]
Hence, this is the 18th term of the given arithmetic progression that is 78.

Note: -The students can make an error in writing the general term for the arithmetic progression by using the formula for finding the \[{{n}^{th}}\] term if they might get confused as the value of ‘n’ is not given and only on proceeding with the question by taking the value of n as an unknown variable, one could get the general term for the arithmetic progression.Remember the general term of arithmetic progression i.e $a,a+d,a+2d,a+3d…...a+(n-1)d$, where a is first term and d is common difference. The general 18th term can be written as $a+17d$, substitute value of a and d we get the required answer i.e $-7+17(5)=78$.