Question
Download Solution PDFThe sum of the first k terms of the arithmetic progression -10, -7, -4, ...., is 104. What is the value of \(\frac{k+9}{k-5}\) ?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFGiven:
Arithmetic progression: -10, -7, -4, ...
Sum of first k terms (Sk) = 104
Formula used:
Sk = (k/2)[2a + (k - 1)d]
where a is the first term and d is the common difference.
Calculation:
a = -10
d = -7 - (-10) = 3
Now, substitute the values into the sum formula:
⇒ 104 = (k/2)[2(-10) + (k - 1)3]
⇒ 104 = (k/2)[-20 + 3k - 3]
⇒ 104 = (k/2)[3k - 23]
⇒ 208 = k(3k - 23)
⇒ 3k2 - 23k - 208 = 0
⇒ 3k2 - 39k + 16k - 208 = 0
⇒ 3k(k - 13) + 16(k - 13) = 0
⇒ (3k + 16)(k - 13)
⇒ k = 13, or -16/3(Negative not possible)
The value of (k + 9) / (k - 5) is
⇒ (13 + 9) / (13 - 5) = 22/8 = 11/4.
Hence the correct answer is 11/4.
Last updated on Apr 24, 2025
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