texelaare

2021-08-15

Arithmetic Sequence? Find the first five terms of the sequence, and determine whether it is arithmetic. If it is arithmetic, find the common difference, and express the nth term of the sequence in the standard form ${a}_{n}=a+\left(n-1\right)d$
${a}_{n}=4+{2}^{n}$

### Answer & Explanation

liannemdh

To find:
The first five terms of the sequence, and check whether it is arithmetic sequence or not. If it is arithmetic sequence, find the common difference, and express the nth term of the sequence in the standard form ${a}_{n}=a+\left(n-1\right)d$
Concept used:
The difference between the successive terms of an arithmetic sequence is constant.
Calculation:
Given sequence is,
${a}_{n}=4+{2}^{n}$ .......(1)
Substitute 1 for n in equation (1) to calculate the first term of this sequence.
${a}_{1}=4+{2}^{1}$
$=4+2$
$=6$
Substitute 2 for n in equation (1) to calculate the second term of this sequence.
${a}_{2}=4+{2}^{2}$
$=4+4$
$=8$
Substitute 3 for n equation (1) to calculate the third term of this sequence.
${a}_{3}=4+{2}^{3}$
$=4+8$
$=12$
Substitute 4 for n equation (1) to calculate the fourth term of this sequence.
${a}_{4}=4+{2}^{4}$
$=4+16$
$=20$
Substitute 5 for n equation (1) to calculate the fifth term of this sequence.
${a}_{5}=4+{2}^{5}$
$=4+32$
$=36$
Difference between first and second term can be calculated as,
${a}_{2}-{a}_{1}=8-6$
$=2$
Difference between second and third term can be calculated as,
${a}_{3}-{a}_{2}=12-8$
$=4$
Since, the difference between the successive terms of the given sequence is not constant.
Therefore, the sequence ${a}_{n}=4+{2}^{n}$ is not arithmetic.
Conclusion:
Hence, the sequence ${a}_{n}=4+{2}^{n}$ is not arithmetic.

Jeffrey Jordon