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Question
The 9th term of an A.P. is 449 and 449th term is 9. The term which is equal to zero is
Options
501th
502th
508th
none of these
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Solution
In the given problem, let us take the first term as a and the common difference as d.
Here, we are given that,
`a_9 = 449` ...............(1)
`a_449 = 9 ` .................(2)
We need to find n
Also, we know,
`a_n = a + ( n- 1) d`
For the 9th term (n = 9),
`a_9 = a + ( 9 -1) d`
449 = a + 8d (Using 1 )
a = 449 - 8 d .................(3)
Similarly, for the 449th term (n = 449),
`a_449 = a + ( 449 - 1 )d`
9 = a + 448d (Using 2 )
a = 9 - 448 d .............(4)
Subtracting (3) from (4), we get,
a -a = ( 9 - 448d) - ( 449 - 8d)
0 = 9 - 448d - 449 + 8d
0 = -440 - 440d
440d = - 440
d = - 1
Now, to find a, we substitute the value of d in (3),
a = 449 - 8 (-1)
a = 449 + 8
a = 457
So, for the given A.P d = - 1 and a = 457
So, let us take the term equal to zero as the nth term. So,
`a_n = 457 + ( n- 1) ( -1 ) `
0 = 457 - n + 1
n = 458
So, n = 458
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