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in Arithmetic Progression by (48.8k points)
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The sum of the first five terms of an AP and the sum of the first seven terms of the same AP is 167. If the sum of the first ten terms of this AP is 235, find the sum of its first twenty terms.

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We know that, in an A.P.,

First term = a

Common difference = d

Number of terms of an AP = n

According to the question,

We have,

S5 + S7 = 167

Using the formula for sum of n terms,

Sn = (n/2) [2a + (n-1)d]

So, we get,

(5/2) [2a + (5-1)d] + (7/2)[2a + (7-1)d] = 167

5(2a + 4d) + 7(2a + 6d) = 334

10a + 20d + 14a + 42d = 334

24a + 62d = 334

12a + 31d = 167

12a = 167 – 31d …(1)

We have,

S10 = 235

(10/2) [2a + (10-1)d] = 235

5[ 2a + 9d] = 235

2a + 9d = 47

Multiplying L.H.S and R.H.S by 6,

We get,

12a + 54d = 282

From equation (1)

167 – 31d + 54d = 282

23d = 282 – 167

23d = 115

d = 5

Substituting the value of d = 5 in equation (1)

12a = 167 – 31(5)

12a = 167 – 155

12a = 12

a = 1

We know that,

S20 = (n/2) [2a + (20 – 1)d]

= 20/(2[2(1) + 19 (5)])

= 10[ 2 + 95]

= 970

Therefore, the sum of first 20 terms is 970.

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