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प्रश्न
The sum of the third and the seventh terms of an AP is 6 and their product is 8. Find the sum of first sixteen terms of the AP.
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उत्तर
We know that,
an = a + (n − 1) d
a3 = a + (3 − 1) d
a3 = a + 2d
Similarly, a7 = a + 6d
Given that, a3 + a7 = 6
(a + 2d) + (a + 6d) = 6
2a + 8d = 6
a + 4d = 3
a = 3 − 4d ...(i)
Also, it is given that (a3) × (a7) = 8
⇒ (a + 2d) × (a + 6d) = 8
From equation (i),
(3 - 4d + 2d) × (3 - 4d + 6d) = 8
(3 - 2d) × (3 + 2d) = 8
9 - 4d2 = 8
4d2 = 9 - 8
4d2 = 1
d2 = `1/4`
d = `± 1/2`
d = `1/2 or 1/2`
From the equation (i)
`("When" "d" 1/2)`
a = 3 - 4d
a = `3 -4(1/4)`
a = 3 - 2
a = 1
When d is − `1/2`
a = `3 - 4(-1/2)`
a = 3 + 2
a = 5
`S_n = n/2 [2a(n-1)s]`
(When a is 1 and d is `1/2`)
`S_16 = (16/2)[2"a" + (16 - 1)"d"]`
`S_16 = 8[2 xx 1 + 15(1/2)]`
`S_16 = 8(2 + 15/2)`
`S_16 = 8 xx 19/2`
S16 = 76
(When a is 5 and d is `-1/2`)
a = `3 - 4(-1/2)`
a = `3 + 4/2`
a = 5
S16 = `(16/2)[2a + (n - 1)d]`
= 8`[2(5) + 15(-1/2)]`
= `8[10 - 15/2]`
= `8 xx 5/2`
= 20
संबंधित प्रश्न
In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato and other potatoes are placed 3 m apart in a straight line. There are ten potatoes in the line.
A competitor starts from the bucket, picks up the nearest potato, runs back with it, drops it in the bucket, runs back to pick up the next potato, runs to the bucket to drop it in, and she continues in the same way until all the potatoes are in the bucket. What is the total distance the competitor has to run?
[Hint: to pick up the first potato and the second potato, the total distance (in metres) run by a competitor is 2 × 5 + 2 ×(5 + 3)]
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Q.4
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