Question

The first term of an AP is 7 and sum of its first 4 terms is half the sum of the next 4 terms. Find the sum of the first 30 terms.

Answer 1

If the 1st term is 7 of an A.P. and the sum of 1st four terms = ½ of sum of next 4 terms, then the sum of the first 30 terms is 1428.

Step-by-step explanation:

Required Formula:

• Sum of n terms in A.P., Sn = $\frac{n}{2} [2a + (n-1)d]$

Where

a = first term of an A.P.

d = common difference  

n = no. of terms in an A.P.

The first term of an A.P. i.e., a = 7

It is given that the sum of the first four terms is half of the sum of the next four terms in an A.P., so we can write the eq. as,

Sum of the first 4 terms = ½ * [{Sum of all the 8 terms} – {sum of first 4 terms}]

⇒ S₄ = ½ * [S₈ – S₄]

Based on the formula and substituting the value of a, we get

⇒ $\frac{4}{2}[(2*7) + (4-1)d] = \frac{8}{2} [(2*7) + (8-1)d] - \frac{4}{2} [(2*7) + (4-1)d]$

⇒ [4{14 + 3d}] = [4{14 + 7d}] – [2{14 + 3d}]

⇒ [4{14 + 3d}] + [2{14 + 3d}] = [4{14 + 7d}]

⇒ [6{14 + 3d}] = [4{14 + 7d}]

⇒ [3{14 + 3d}] = [2{14 + 7d}]

⇒ 42 + 9d = 28 + 14d

⇒ 5d = 14

⇒ d = $\frac{14}{5}$

⇒ d = 2.8

Thus,

The sum of the 30 terms is given by,

= S₃₀

= $\frac{30}{2} [(2*7) + (30-1)2.8]$

= [15 * { 14 + (29 * 2.8)}]

= [15 * {14 + 81.2}]

= 15 * 95.2

= 1428

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Also View:

Find the common difference of an A.P. whose first term is 5 and the sum of first four terms is half the sum of next four terms.

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If the first term of an A.P. is 2 and common difference is 4, then the sum of its 40 terms is

(a) 3200

(b) 1600

(c) 200

(d) 2800

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