If the following is an arithmetic sequence, then find the value of \(n\) :
\(\{-1,n,13…\}\)
Detailed look at the solution:
We are given that \(\{-1,{\color{red}{n}},13,…\}\) is an arithmetic sequence.
An arithmetic sequence is a list of numbers in which each term is obtained by adding a fixed number to the preceding term, except the first
term.
If we assume this fixed number to be \(d\) in the given sequence, this means the following:
\(-1+d=n~ ~ —(1)\)
\(n+d=13~ ~ —(2)\)
\((2)\implies d = 13 – n~ ~ —(3)\)
Substituting this value of \(d\) in \((1),\) we get:
\(-1+(13-n)=n\implies 12-n=n\implies 12=2n\implies n=\frac{12}{2}=6\)
Thus, the value of \(n\) equals \(6.\)
Verification:
Consider \(\{-1,{\color{red}{6}},13\}.\)
\(6-(-1)=6+1=7\)
\(13-6 = 7\)
\(\therefore 6-(-1)=13-6=7,\) a constant.
Thus, we have \(n=6.\)