Question

How do you find the formula for ${a_n}$ for the arithmetic sequence when ${a_1} = - 4$ and ${a_5} = 16$?

Answer 1

A series of numbers such that the difference between consecutive terms is equal or constant is an arithmetic progression (AP) or arithmetic sequence. For any arithmetic series, the general formula is written as – the nth term is equal to the first term plus the general difference times the number of differences between the first term and the nth term.
${a_n} = a + (n - 1)d$
Here ${a_n}$ is the nth term, $a$ is the first term and $d$ is the common difference between any two consecutive numbers in the series. We can put the value of the variables in the above equation to find the value of common difference $d$. Then we will put the values of $a$ and $d$ in the general equation to get the required formula.

Complete step by step solution:
The question mentions that we have an arithmetic sequence. In an arithmetic sequence, there is a starting term or the first term and the difference between any two consecutive terms of an arithmetic sequence is equal or constant.
The general equation of an arithmetic sequence is given by:
${a_n} = a + (n - 1)d$
Now we put the value of the variables whose values are known.
Here, the first term ${a_1} = a = - 4$ and ${a_5} = 16$. Also, $n = 5$.

$$\Rightarrow {a_5} = a + (5 - 1)d \\\ \Rightarrow 16 = - 4 + 4d \\\ \Rightarrow 4d = 16 + 4 \\\ \Rightarrow 4d = 20 \\\ \Rightarrow d = \dfrac{{20}}{4} = 5 \\\$$

Hence, the common difference $d = 5$.
Now, we put the values of $a$ and $d$ in the general equation.

$${a_n} = a + (n - 1)d \\\ \Rightarrow {a_n} = - 4 + (n - 1) \times 5 \\\ \Rightarrow {a_n} = - 4 + 5n - 5 \\\ \Rightarrow {a_n} = 5n - 9 \\\$$

**Hence, ${a_n} = 5n - 9$ is the required general formula when ${a_1} = - 4$ and ${a_5} = 16$
**

Note: In case of arithmetic series, each succeeding number is some value (known as the common difference) more than the preceding number. The common difference can be negative or positive, depending on the case. Arithmetic series should not be confused with Geometric series as in Geometric series, each succeeding number is some value (known as the common ratio) multiplied by the preceding number.