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in Arithmetic Progression by (56.3k points)

In an A.P., if the 5th and 12th terms are 30 and 65 respectively, what is the sum of first 20 terms?

1 Answer

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Best answer

Let’s take the first term as a and the common difference to be d

Given that,

a5 = 30  and a12 = 65

And, we know that an = a + (n – 1)d

So,

a5 = a + (5 – 1)d

30 = a + 4d

a = 30 – 4d   …. (i)

Similarly, a12 = a + (12 – 1) d

65 = a + 11d

a = 65 – 11d …. (ii)

Subtracting (i) from (ii), we have

a – a = (65 – 11d) – (30 – 4d)

0 = 65 – 11d – 30 + 4d

0 = 35 – 7d

7d = 35

d = 5

Putting d in (i), we get

a = 30 – 4(5)

a = 30 – 20

a = 10

Thus for the A.P; d = 5 and a = 10

Next, to find the sum of first 20 terms of this A.P., we use the following formula for the sum of n terms of an A.P.,

Sn =\(\frac{ n}{2}\)[2a + (n − 1)d]

Where;

a = first term of the given A.P.

d = common difference of the given A.P.

n = number of terms

Here n = 20, so we have

S20 = \(\frac{20}{2}\)[2(10) + (20 − 1)(5)]

= (10)[20 + (19)(5)]

= (10)[20 + 95]

= (10)[115]

= 1150

Hence, the sum of first 20 terms for the given A.P. is 1150.

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