Answer :

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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