In an A.P. 19th term is 52 and 38th term is 128.
Find the sum of first 56 terms.
Answers
Step-by-step explanation:
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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