# The sum of 16th term of an Ap is 240 and the next 4 terms is 220. Find the first term, common difference.

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1. If your question means:

The sum of 16th term of an Ap is 240 and the sum of next 4 terms is 220.

then

Sum of first n terms of an AP:

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

In this case n = 16.

S16 = 240

16 ( 2 a + 15 d ) / 2 = 240

Multiply both sides by 2

16 ( 2 a + 15 d ) = 480

Divide both sides by 16

2 a + 15 d = 30

In AP

an = a + ( n - 1 ) d

a17 = a + 16 d

a18 = a + 17 d

a19 = a + 18 d

a20 = a + 19 d

The sum of next 4 terms:

a17 + a18 + a19 + a20 = 220

a + 16 d + a + 17 d + a + 18 d + a + 19 d = 220

4 a + 70 d = 220

Now you must solve system:

2 a + 15 d = 30

4 a + 70 d = 220

The solution is:

a = - 15 , d = 4

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2. Of course:

a = first term

d = common difference.

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3. sum of 16 terms is 240
8(2a + 15d) = 240
2a + 15d = 30

sum of 20 terms is 460 , (the first 240 plus the next 4 of 220 )
10(2a + 19d) = 460
2a + 19d = 46

subtract them: 4d = 16
d = 4
sub into 2a+15d = 30
2a + 60 = 30
a = -15

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4. Yes

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