In a decreasing arithmetic progression, the sum of all its terms, except the first term, is equal to -75. The sum of all its terms except the last term is zero and the difference of the 9th term and the 13th term is equal to -20. What is the first term of this series.
Explanation:
Let the sum of all the terms of the series = S.
∴ S = T1 + (-75) ⇒ T1 = S + 75
Similarly, S = 0 + Tn ⇒ Tn = S
T13 - T9 = - 20 ⇒ (T1 + 12d) - (T1 + 8d) = - 20 ⇒ 4d = - 20 ⇒ d = - 5
Now, Tn = T1 + (n - 1)d ⇒ S = (S + 75) + -5(n - 1) ⇒ -75 = -5(n - 1) ⇒ 15 = (n - 1) ⇒ n = 16
Now, S = n/2 × [T1 + Tn] ⇒ S = 16/2 × [S + 75 + S] ⇒ S = 8[2S + 75] ⇒ S = 16S + 8 × 75 ⇒ 15S = - 8 × 75 ⇒ S = -40
Finally, T1 = S + 75 = - 40 + 75 = 35
Hence, option (b).
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