Question

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Answer 1

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Answer 2

Question:

In an A.P , if S₅ + S₇ = 167 and S₁₀ = 235 . Then find the AP where Sₙ denotes the sum of first n terms .

Answer:

1 , 6 , 11 ...

Step-by-step explanation:

Let the first term of the AP be a .

Let the common difference of the AP be d .

We know that the sum of n terms is given by the formula :

Sn = n/2 [ 2 a + ( n - 1 ) d .

Hence sum of 10 terms will be :

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

⇒ 235 = 5 [ 2 a + 9 d ]

⇒ 2 a + 9 d = 235/5

⇒ 2 a + 9 d = 47 --------(1)

S₅ + S₇ = 167

⇒ 5/2 [ 2 a + 4 d ] + 7/2 [ 2 a + 6 d ] = 167

⇒ 5/2 × 2 ( a + 2 d ) + 7/2 × 2 ( a + 3 d ) = 167

⇒ 5 ( a + 2 d ) + 7 ( a + 3 d ) = 167

⇒ 5 a + 10 d + 7 a + 21 d = 167

⇒ 12 a + 31 d = 167 -------(2)

Multiplying equation (1) by 6 we get :

⇒ 12 a + 54 d = 282 -----(3)

Subtracting equation (3) from (2) we get :

⇒ 54 d - 31 d = 282 - 167

⇒ 23 d = 115

⇒ d = 115/23

⇒ d = 5

Putting the value of d in equation 1 we get :

⇒ 2 a + 9(5) = 47

⇒ 2 a + 45 = 47

⇒ 2 a = 47 - 45

⇒ 2 a = 2

⇒ a = 2/2

⇒ a = 1

A.P = 1 , 1 + 5 , 1 + 10

⇒ 1 , 6 , 11 ......