Question
Show that the sequence defined by $a_n = 5n - 7$ is an A.P, find its common difference.
✓
Answer
In the given problem, we need to show that the given sequence is an A.P. and then find its common difference.
Here,
$a_n = 5n - 7$
Now, to show that it is an A.P, Will find its few terms by substituting $n = 1, 2, 3, 4, 5$
So,
Substituting n = 1, we get
$a_1 = 5(1) - 7$
$a_1 = -2$
Subsituting n = 2, we get
$a_2 = 5(2) - 7$
$a_2 = 3$
Subsituting n = 3, we get
$a_3 = 5(3) - 7$
$a_3 = 8$
Subsituting n = 4, we get
$a_4 = 5(4) - 7$
$a_4 = 13$
Subsituting n = 5, we get
$a_5 = 5(5) - 7$
$a_5 = 18$
Further, for the given sequence to be an A.P,
We find the common difference (d)
$a = a_2 - a_1 = a_3 - a_2$
thus,
$a_2 - a_1 = 3 - (-2) = 5$
Since $a_2 - a_1 = a_3 - a_2$
Hence, the given sequence is an A.P, and its common difference is d = 5.
Here,
$a_n = 5n - 7$
Now, to show that it is an A.P, Will find its few terms by substituting $n = 1, 2, 3, 4, 5$
So,
Substituting n = 1, we get
$a_1 = 5(1) - 7$
$a_1 = -2$
Subsituting n = 2, we get
$a_2 = 5(2) - 7$
$a_2 = 3$
Subsituting n = 3, we get
$a_3 = 5(3) - 7$
$a_3 = 8$
Subsituting n = 4, we get
$a_4 = 5(4) - 7$
$a_4 = 13$
Subsituting n = 5, we get
$a_5 = 5(5) - 7$
$a_5 = 18$
Further, for the given sequence to be an A.P,
We find the common difference (d)
$a = a_2 - a_1 = a_3 - a_2$
thus,
$a_2 - a_1 = 3 - (-2) = 5$
Since $a_2 - a_1 = a_3 - a_2$
Hence, the given sequence is an A.P, and its common difference is d = 5.
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