Question

How do you write the first five terms of the arithmetic sequence given ${{a}_{1}}=15,{{a}_{k+1}}={{a}_{k}}+4$ and find the common difference and write the ${{n}^{th}}$ term of the sequence as a function of n?

Answer 1

In this problem, we have to find the first five terms of the arithmetic sequence and common difference from the given data. We can first write the general form of the arithmetic sequence. We can then substitute k values in the given condition to find the sequence.

Complete step by step solution:
We know that the given data are,
${{a}_{1}}=15,{{a}_{k+1}}={{a}_{k}}+4$
Here the first term is ${{a}_{1}}$.
We know that the general form of the arithmetic sequence is,
${{a}_{n}}={{a}_{1}}+\left( n-1 \right)d$
Where d is the common difference.
We can now assume values for k, and substitute it in ${{a}_{k+1}}={{a}_{k}}+4$.
We can now take k = 1, we get

& \Rightarrow {{a}_{1+1}}={{a}_{1}}+4 \\\ & \Rightarrow {{a}_{2}}=15+4=19 \\\ \end{aligned}$$ We can now take k = 2, we get $$\begin{aligned} & \Rightarrow {{a}_{2+1}}={{a}_{2}}+4 \\\ & \Rightarrow {{a}_{3}}=19+4=23 \\\ \end{aligned}$$ We can now take k = 3, we get $$\begin{aligned} & \Rightarrow {{a}_{3+1}}={{a}_{3}}+4 \\\ & \Rightarrow {{a}_{4}}=23+4=27 \\\ \end{aligned}$$ We can now take k = 4, we get $$\begin{aligned} & \Rightarrow {{a}_{4+1}}={{a}_{4}}+4 \\\ & \Rightarrow {{a}_{5}}=27+4=31 \\\ \end{aligned}$$ Therefore, first five terms of the arithmetic sequence; $$15,19,23,27,31$$ We can now find the common difference by subtracting the term with its previous term. $$d=19-15=23-19=27-23=4$$ Therefore, the common difference, d = 4. We can now write the $${{n}^{th}}$$term of the sequence, $${{a}_{n}}={{a}_{1}}+\left( n-1 \right)4=15+4\left( n-1 \right)$$ Therefore, the first five terms of the arithmetic sequence; $$15,19,23,27,31$$. the common difference, d = 4 and the $${{n}^{th}}$$term of the sequence, $${{a}_{n}}={{a}_{1}}+\left( n-1 \right)4=15+4\left( n-1 \right)$$ **Note:** Students make mistakes while finding the consecutive terms of the arithmetic sequence by substituting value to the unknown variable. We can find the common difference by subtracting the term from its previous term. We should also remember the general formula for the arithmetic sequence.