# In an arithmetic progression’s the sum of the first fifteen terms is 24/65 times of the sum of the first twenty-five terms. If the first term of the A

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In an arithmetic progression’s the sum of the first fifteen terms is 24/65 times of the sum of the first twenty-five terms. If the first term of the A.P is -2, then, find the sum of the first 30 terms?
1. -920
2. -950
3. -840
4. -930

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Correct Answer - Option 4 : -930

Given:

a = -2,

S15 = 24/65 × S25

Formula used:

Sn = n/2 × [2a + (n – 1) d]

Where, a = first term, n = number of terms and d = common difference

Calculation:

S15 = 15/2 × [2(-2) + (15 – 1) d]

⇒ S15 = 15/2 × (- 4 + 14d)

⇒ S15 = (-30 + 105d)     ---- (1)

S25 = 25/2 × [2(-2) + (25 – 1) d]

⇒ S25 = 25/2 × (-4 + 24d)

⇒ S25 = (-50 + 300d)      ----- (2)

According to question –

⇒ S15 = 24/65 × S25

⇒ -30 + 105d = 24/65 × (-50 + 300d)

⇒ -1950 + 6825d = -1200 + 7200d

⇒ (7200 – 6825) d = (-1950 + 1200)

⇒ 375d = -750

d = -2

Now, S30 = 30/2 × [2(-2) + (30 – 1) × (-2) ]

⇒ S30 = 15[- 4 - 58]

⇒ S30 = 15 × (-62)

∴ S30 = - 930