Given: Total debt is Rs. 3600 and total number of installments are 40 and in 30 installments, he has paid two-third of the debt and dies leaving one-third of amount

Let first instalment be “a”

Let the common difference of the instalments be “d”

Here, S₄₀ = 3600 and n = 40

Formula used:

where a is first term, d is common difference and n is number of terms in an A.P.

Therefore,

⇒ 2a+39d = 180 ………(1)

Since 30 installments are paid and one third of the debt unpaid

So,

S₃₀ = 2400 and n = 30

⇒ 2a+29d = 160 ………(2)

Solving (1) and(2) by substitution method,

2a + 39d = 180

⇒ 2a = 180 – 39d ……(3)

Put value of 2a from (3) in (2):

2a + 29d = 160

⇒ 180 – 39d + 29d = 160

⇒ –10d = 160 – 180

⇒ –10d = –20

⇒ d = 2

Put this value of d in (3):

2a = 180 – 39d

⇒ 2a = 180 – 39(2)

⇒ 2a = 180 – 78

⇒ 2a = 102

⇒ a = 51

Hence, value of first installment = a = Rs. 51