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Question
Find the amount accumulated after 2 years if a sum of ₹ 24,000 is invested every six months at 12% p.a. compounded half yearly. [Given (1.06)4 = 1.2625]
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Solution
Given, C = ₹ 24,000, Since amount is invested at the end of every 6 months for two years, it is an immediate annuity.
∴ n = 2 x 2 = 4 half years.
Rate of interest is 12% p.a
∴ r = `(12)/(2)` = 6% for six months
i = `"r"/(100) = (6)/(100)` = 0.06
Now, A = `"C"/"i"[(1 + "i")^"n" - 1]`
= `(24,000)/(0.06)[(1 + 0.06)^4 - 1]`
= `(24,000 xx 100)/(0.06 xx 100)[(1.06)^4 - 1`
= `(24,00,000)/6(1.2625 - 1)`
= 4,00,000 × 0.2625
∴ = 1,05,000
∴ Amount accumulated after 2 years is ₹ 1,05,000.
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[Given (1.1)4 = 1.4641]
For annuity due,
C = ₹ 20,000, n = 3, I = 0.1, (1.1)–3 = 0.7513
Therefore, P = `square/0.1 xx [1 - (1 + 0.1)^square]`
= 2,00,000 [1 – 0.7513]
= ₹ `square`
