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प्रश्न
Find the second term and nth term of an A.P. whose 6th term is 12 and 8th term is 22.
उत्तर
In the given problem, we are given 6th and 8th term of an A.P.
We need to find the 2nd and nth term
Here, let us take the first term as a and the common difference as d
We are given,
`a_6 = 12`
`a_8 = 22`
Now we will find `a_6` and `a_8` using the formula `a_n = a + (n - 1)d`
So
`a_6 = a + (6 -1)d`
12 = a + 5d ....(1)
Also
`a_8 = a + (8 -1)d`
22 = a + 7d ....(2)
So to solve for a and d
On subtracting (1) from (2), we get
22 - 12 = (a + 7d) - (a + 5d)
10 = a + 7d - a - 5d
10 = 2d
`d = 10/2`
d = 5 ....(3)
Substituting (3) in (1) we get
12 = a + 5(5)
a = 12 - 25
a = -13
Thus
a= -13
d = 5
So, for the 2 nd term (n = 2)
`a_2 = -13 + (2 - 1)5`
= -13 + (1)5
= -13 + 5
= -8
For the nth term
`a_n = -13 + (n - 1)5`
= -13 + 5n - 5
= -18 + 5n
Therefore `a_2 = -8, a_n = 5n - 18`
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