What is the value of 'k' in the following sum ? 57 + 54 + 51+ ........+ k = -123

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14 Jul 2022 Sequence & Series

What is the value of 'k' in the following sum ?
57 + 54 + 51+ ........+ k = -123

OPtion

1) 60
2) 63
3) -60
4) -51
5) -75
6) -63
7) -48
8) -54
9) -66
10) None of these

Solution

Given, arithmetic sequence 57, 54, 51, ....., k. Here first term a=57, Difference d=-3, Last term=k, Sum of all terms=-123
Sum of the n terms of an A.P. = (n/2)[2a + (n-1)d]
-123 = (n/2)[2*57 + (n-1)*(-3)]
⇒ n² - 39n - 82 = 0
(n-41)(n+2) = 0
Considering positive value of number of terms, n=41 ⇒ k=41th term of the series.
Now, n th term of an AP = a + (n-1)d
.'. k = 57 + 40*(-3) = -63

Correct Option: 6 Best Solution (0)