Question

Given the first term and the common difference of an arithmetic sequence how do you find the first five terms and explicit formula : ${{a}_{1}}=28,d=10?$

Answer 1

Hint: To find the solution of the problem first we need to know about arithmetic sequence, by applying the general formula ${{a}_{n}}={{a}_{1}}+(n-1)d$ and here in this problem we have given values ${{a}_{1}}=28$ and $d=10$. By applying this in the general equation we can find the first five values of the given arithmetic sequence.

Complete step-by-step answer:
 From the question we know, ${{a}_{1}}=28$ is first term of the series and $d=10$is the common difference of the series given, by applying the general formula we get,
$\begin{align}
  & {{a}_{2}}={{a}_{1}}+d=28+10 \\
 & \Rightarrow {{a}_{2}}=38 \\
\end{align}$ Similarly,
$\begin{align}
  & \Rightarrow {{a}_{3}}={{a}_{2}}+d=38+10=48 \\
 & \Rightarrow {{a}_{4}}={{a}_{3}}+d=48+10=58 \\
 & \Rightarrow {{a}_{5}}={{a}_{4}}+d=58+10=68 \\
\end{align}$
Here, we get all the first five terms of the given arithmetic series those are
${{a}_{1}}=28,{{a}_{2}}=38,{{a}_{3}}=48,{{a}_{4}}=58,{{a}_{5}}=68$
The explicit formula of the given series is also calculated by the same general formula given in the solution of the problem,
$\begin{align}
  & {{a}_{n}}={{a}_{1}}+(n-1)d \\
 & \Rightarrow {{a}_{n}}=28+(n-1)10 \\
 & \Rightarrow {{a}_{n}}=28+10n-10 \\
 & \Rightarrow {{a}_{n}}=18+10n \\
\end{align}$
The resultant ${{a}_{n}}=18+10n$ is the explicit formula of the given arithmetic sequence.

Note: To solve the particular given problem, one should have the basic knowledge of arithmetic sequence and should know about the terms and the meaning of difference in the series, one should aware of the formula that is used to find the terms in the given problem, without that particular formula one cannot get the particular or accurate solution for the given arithmetic sequence problem, one may go wrong in adding and subtracting if the difference value is negative, in this given problem it is positive value, so it makes us easy to understand and solve the given problem.