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The ratio of the 8th and 17th terms of an AP is 5 : 11. The product of the first and the sixth terms of the AP is 176. If the AP has only negative terms in the series, find the sum of the first 200 terms of the series.
  • a)
    -160000
  • b)
    -162000
  • c)
    -158000
  • d)
    -80000
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
The ratio of the 8th and 17th terms of an AP is 5 : 11. The product of...
Given:
- The ratio of the 8th term to the 17th term of an arithmetic progression (AP) is 5:11.
- The product of the first and sixth terms of the AP is 176.
- The AP consists of only negative terms.

To find:
The sum of the first 200 terms of the AP.

Solution:
Let's assume the first term of the AP is 'a' and the common difference is 'd'.

Ratio of 8th and 17th terms:
The 8th term of the AP can be expressed as:
a + 7d

The 17th term of the AP can be expressed as:
a + 16d

Given that the ratio of the 8th and 17th terms is 5:11, we have the equation:
(a + 7d) / (a + 16d) = 5/11

Cross-multiplying, we get:
11(a + 7d) = 5(a + 16d)

Simplifying further, we get:
11a + 77d = 5a + 80d
6a = 3d
2a = d

Product of first and sixth terms:
The first term of the AP is 'a'.

The sixth term of the AP can be expressed as:
a + 5d

Given that the product of the first and sixth terms is 176, we have the equation:
a(a + 5d) = 176

Substituting the value of 'd' from the previous equation, we get:
a(a + 10a) = 176
11a^2 = 176
a^2 = 16
a = -4 (since the AP consists of only negative terms)

Sum of the first 200 terms:
The sum of an arithmetic progression can be calculated using the formula:
Sum = (n/2)(2a + (n-1)d)

We are given that the AP consists of negative terms, so the sum will also be negative.

Substituting the values in the formula, we get:
Sum = (200/2)(2(-4) + (200-1)(2))
Sum = -100(-8 + 398)
Sum = -100(390)
Sum = -39,000

Therefore, the sum of the first 200 terms of the series is -39,000, which is equivalent to option A.
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Community Answer
The ratio of the 8th and 17th terms of an AP is 5 : 11. The product of...
Let the first term be a and the common difference be d. We have, (a + 7d)/(a + 16d) = 5/11.
Solving this, we get d = 2a. Product of the first and sixth terms is a (a + 5d) = a(a + 10 a) = 11a2 = 176 ⇒ a = + - 4
Since the series has only negative terms, a=-4 and d=-8.
Sum of the first 200 terms = (200/2)(2*-4 + 199*- 8) = -160000
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The ratio of the 8th and 17th terms of an AP is 5 : 11. The product of the first and the sixth terms of the AP is 176. If the AP has only negative terms in the series, find the sum of the first 200 terms of the series.a)-160000b)-162000c)-158000d)-80000Correct answer is option 'A'. Can you explain this answer?
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