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
Now we have to find the value of “a” and “d”
Using n^th term of an A.P. formula t_n = a + (n – 1) d
where n = no. of terms
a = first term
d = common difference
t_n = n^th 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 equation (1) from equation (2), we get,
⇒ 128 – 52 = (a – a) + (37 d – 18 d)
⇒ 76 = 19 d
⇒ d = 76/19 = 4
Substitute value of “d” in equation (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
S_n = sum of n terms
Thus, substituting given value in formula we can find the value of S_n

⇒S₅₆ = 28 × [ – 40 + 55×4]
⇒S₅₆ = 28 × [ – 40 + 220]
⇒S₅₆ = 28 × 180 = 5040
Thus, S₅₆ = 5040