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प्रश्न
Show that the sequence defined by an = 5n −7 is an A.P, find its common difference.
उत्तर
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, we 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`
Substituting n = 2, we get
`a_2 = 5(2) - 7`
`a_2 = 3`
Substituting n = 3, we get
`a_3 = 5(3) - 7`
`a_3 = 8`
Substituting n = 4, we get
`a_4 = 5(4) - 7`
`a_4 = 13`
Substituting 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
Also
`a_3 - a_2 = 8 - 3`
= 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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