# In an A.P. 19th term is 52 and 38th term is 128, find sum of first 56 terms.

Given: t19 = 52 and t38 = 128

To find: value of “a” and “d”

Using nth term of an A.P. formula

tn = a + (n – 1)d

where n = no. of terms

a = first term

d = common difference

tn = nth terms

we will find value of “a” and “d”

Let, t19 = a + (19 – 1) d

52 = a + 18 d …..(1)

t38 = a + (38 – 1) d

128 = a + 37 d …..(2)

Subtracting eq. (1) from eq. (2), we get,

128 – 52 = (a – a) + (37 d – 18 d)

76 = 19 d

Substitute value of “d” in eq. (1) to get value of “a”

52 = a + 18 ×4

52 = a + 72

a = 52 – 72 = – 20

Now, to find value of S56 we will using formula of sum of n terms

Where, n = no. of terms

a = first term

d = common difference

Sn = sum of n terms

Thus, Substituting given value in formula we can find the value of Sn

S56 = 28 × [ – 40 + 55×4]

S56 = 28 × [ – 40 + 220]

S56 = 28 × 180 = 5040

Thus, S56 = 5040

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