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Question
If the nth term of the A.P. 9, 7, 5, ... is same as the nth term of the A.P. 15, 12, 9, ... find n.
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Solution
Here, we are given two A.P. sequences whose nth terms are equal. We need to find n.
So let us first find the nth term for both the A.P.
First A.P. is 9, 7, 5 …
Here
First term (a) = 9
Common difference of the A.P. (d) = 7 - 9
= -2
Now as we know
`a_n = a + (n - 1)d`
So for nth term
`a_n = a + (n -1)d`
So for nth term
`a_n = 9 + (n -1)(-2)`
= 9 - 2n + 2
= 11 - 2n .......(1)
Second A.P. is 15, 12, 9 …
Here,
First term (a) = 15
Common difference of the A.P. (d) = 12 - 15
= -3
Now as we know
`a_n = a + (n - 1)d`
So for nth term
`a_n = 15 + (n -1)(-3)`
= 15 - 3n + 3
= 18 - 3n .....(2)
Now, we are given that the nth terms for both the A.P. sequences are equal, we equate (1) and (2),
11 - 2n = 18 - 3n
3n - 2n = 18 - 11
n = 7
Therefore n = 7
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