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The sum of a certain number of terms of an AP series 8, 6, 4, …… is -52. The number of terms is
- a)12
- b)13
- c)11
- d)none of these
Correct answer is option 'B'. Can you explain this answer?
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The sum of a certain number of terms of an AP series 8, 6, 4, …...
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The sum of a certain number of terms of an AP series 8, 6, 4, …...
Given: AP series 8, 6, 4; Sum of terms = 52
To find: Number of terms in the series
Approach:
Let's assume that the AP series has 'n' terms.
We need to find the value of 'n'.
Formula:
The sum of 'n' terms of an AP series can be calculated using the formula -
Sn = (n/2) * [2a + (n-1)d]
Where,
Sn = Sum of 'n' terms
a = First term of the series
d = Common difference between the terms
Calculation:
Given,
a = 8 (first term)
d = 6 - 8 = -2 (common difference)
Sn = 52
Using the formula,
52 = (n/2) * [2(8) + (n-1)(-2)]
52 = (n/2) * [16 - 2(n-1)]
52 = (n/2) * (18 - 2n)
104 = n(18 - 2n)
2n^2 - 18n + 104 = 0
n^2 - 9n + 52 = 0
Solving the quadratic equation, we get -
n = 4 or n = 13
The number of terms in the series cannot be 4 as the given series has more than 4 terms.
Therefore, the number of terms in the series is 13.
Hence, the correct option is (B).
To find: Number of terms in the series
Approach:
Let's assume that the AP series has 'n' terms.
We need to find the value of 'n'.
Formula:
The sum of 'n' terms of an AP series can be calculated using the formula -
Sn = (n/2) * [2a + (n-1)d]
Where,
Sn = Sum of 'n' terms
a = First term of the series
d = Common difference between the terms
Calculation:
Given,
a = 8 (first term)
d = 6 - 8 = -2 (common difference)
Sn = 52
Using the formula,
52 = (n/2) * [2(8) + (n-1)(-2)]
52 = (n/2) * [16 - 2(n-1)]
52 = (n/2) * (18 - 2n)
104 = n(18 - 2n)
2n^2 - 18n + 104 = 0
n^2 - 9n + 52 = 0
Solving the quadratic equation, we get -
n = 4 or n = 13
The number of terms in the series cannot be 4 as the given series has more than 4 terms.
Therefore, the number of terms in the series is 13.
Hence, the correct option is (B).
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Community Answer
The sum of a certain number of terms of an AP series 8, 6, 4, …...
Formula is n/2(2a+(n-1)d)
so n/2(2(-8)+(n-1)2)=n/2(-16+2n-2)
=n/2(-18+2n)=52
-18n+2n^2=104
n^2-9n-52=0
n^2-13n+4n-52=0
n(n-13)+4(n-13)=0
(n-13)(n+4)=0
n=13. n=-4
Here we consider only positive values so n=13
Hence it is proved
so n/2(2(-8)+(n-1)2)=n/2(-16+2n-2)
=n/2(-18+2n)=52
-18n+2n^2=104
n^2-9n-52=0
n^2-13n+4n-52=0
n(n-13)+4(n-13)=0
(n-13)(n+4)=0
n=13. n=-4
Here we consider only positive values so n=13
Hence it is proved
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The sum of a certain number of terms of an AP series 8, 6, 4, …… is -52. The number of terms isa)12b)13c)11d)none of theseCorrect answer is option 'B'. Can you explain this answer?
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