Question

How do you write the first five terms of the arithmetic sequence, given ${a_1} = 5,$ $d = 6?$

Answer 1

When the first term and the common difference d is given, we can find the next term by adding the common difference d to the preceding number. Here we use the formula ${a_{n + 1}} = {a_n} + d$.
In this problem we need only the first five terms, we put $n = 1,2,3,4$. Then we obtain a sequence of the form ${a_1},{a_2},{a_3},{a_4},{a_5}$ which are the first five terms of an arithmetic sequence.

Complete step by step solution:
Given ${a_1} = 5,$ $d = 6.$
We need to find the first five terms of the arithmetic sequence.
Generally we represent an arithmetic sequence by $a,a + d,a + 2d,a + 3d,.....$
where ${a_1} = a,$ ${a_2} = a + d,$ ${a_3} = a + 2d,$ ${a_4} = a + 3d$ and so on.
Where, $a =$ First term of arithmetic progression
$d =$ Common difference
In this problem, we find the first five terms of an arithmetic sequence by using the formula,
${a_{n + 1}} = {a_n} + d$ ……(1)
for $n = 1,2,3,4$.
Here we note that ${a_1} = a$ which is the required first term of an arithmetic sequence.
To obtain the second term, we put $n = 1$ in the equation (1), we get,
${a_{1 + 1}} = {a_1} + d$
Substituting ${a_1} = 5$ and $d = 6$, we get,
${a_2} = 5 + 6$
$\Rightarrow {a_2} = 11.$
To obtain the third term, we put $n = 2$ in the equation (1), we get,
${a_{2 + 1}} = {a_2} + d$
Substituting the values of ${a_2} = 11$ and $d = 6$, we get,
${a_3} = 11 + 6$
$\Rightarrow {a_3} = 17.$
To obtain the fourth term, we put $n = 3$ in the equation (1), we get,
${a_{3 + 1}} = {a_3} + d$
Substituting the values of ${a_3} = 17$and $d = 6$, we get,
${a_4} = 17 + 6$
$\Rightarrow {a_4} = 23.$
To obtain the fifth term, we put $n = 4$ in the equation (1), we get,
${a_{4 + 1}} = {a_4} + d$
Substituting the values of ${a_4} = 23$ and $d = 6$, we get,
${a_5} = 23 + 6$
$\Rightarrow {a_4} = 29.$
$\therefore {a_1} = 5,$ ${a_2} = 11,$ ${a_3} = 17,$ ${a_4} = 23,$ ${a_5} = 29.$
Hence the first five terms of an arithmetic sequence when ${a_1} = 5,$ $d = 6$ is given by,
$5,11,17,23,29.$

Note:
An arithmetic progression is a list of numbers in which each term is got by adding a fixed number to the preceding term except the first term.
The fixed number is called the common difference of the arithmetic progression and it is denoted by d. Note that the common difference d can be positive, negative or zero.
If ${a_1},{a_2},{a_3},{a_4},{a_5},....,{a_{n - 1}},{a_n}$ is an arithmetic progression, then we note that
$d = {a_2} - {a_1} = {a_3} - {a_2} = {a_4} - {a_3} = .... = {a_n} - {a_{n - 1}}.$
In order to determine whether a list of numbers is an arithmetic progression or not we need to just find the common difference d between the two numbers. If d is constant for all the terms then it is an arithmetic sequence. Otherwise it is not an arithmetic sequence.