What sum of money should be deposited at the end of every 6 months to accumulate ₹ 50,000 in 8 years, if money is worth 6% p.a. compounded semi-annually ? [Given : $(1.03)^{16} = 1.6047$] (A) ₹ 3,432.53 (B) ₹ 2,783.08 (C) ₹ 2,480.57 (D) ₹ 2,149.93

> [!TLDR] > Using the future-value-of-annuity formula with $i=0.03,\,n=16$, the half-yearly deposit is about ₹2,480.57. > [!FORMULA] > Future value of an ordinary annuity: $FV=R\cdot\dfrac{(1+i)^{n}-1}{i}$, where $R$ is each deposit, $i$ the rate per period, $n$ the number of periods. 1. Semi-annual rate $i=\dfrac{6\%}{2}=0.03$; number of periods $n=8\times2=16$. 2. $(1+i)^{16}=(1.03)^{16}=1.6047$ (given). …

What sum of money should be deposited at the end of every 6 months to accumulate ₹ 50,000 in 8 years, if money

Question

(1.03)^{16} = 1.6047

CBSECBSE Class XII Board 2024MCQ· 1mImportance★★★★★

🔒 Locked · start free trial →

You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.

Start your 14-day free trial to unlock the full solution →

Concept understanding — Present Value Perpetuity

Present Value of a Perpetuity: The Intuition

Imagine you own a magic tree that gives you exactly ₹100 every year, forever. No matter what happens, that ₹100 arrives each year, year after year, for eternity. What would you sell that tree for today? That's the question the present value of a perpetuity answers.

The key insight is simple: a rupee today is worth more than a rupee tomorrow. So those future ₹100 payments, when brought back to today, are worth less than ₹100 each. The further away the payment, the less it's worth today.

If you think about it, the first ₹100 (one year from now) is worth about ₹91 today (at 10% interest). The second ₹100 (two years from now) is worth about ₹83 today. The third is worth about ₹75. And so on, getting smaller and smaller.

Now here's the beautiful part: if you add up all these shrinking values — the ₹91, the ₹83, the ₹75, and so on forever — they don't add up to infinity. They converge to a finite number. That finite number is the present value of the perpetuity.

Note

A perpetuity is just an annuity that never ends. An annuity pays you for a fixed number of years; a perpetuity pays you forever.

The Precise Statement

Let:

$C$ = the constant cash flow received each period (e.g., ₹100 per year)
$r$ = the discount rate per period (e.g., 10% = 0.10)

The present value $PV$ of a perpetuity is:

PV = \frac{C}{r}

That's it. One division. No summation, no infinite series.

For our example: $PV = \frac{100}{0.10} = ₹1000$ .

So the magic tree is worth ₹1000 today. Why? Because if you put ₹1000 in a bank account earning 10% per year, you'd get ₹100 in interest each year — forever — without ever touching the principal. The tree and the bank account produce the same cash flow, so they have the same value.

Where Does This Formula Come From?

The present value is the sum of all future cash flows discounted back to today:

PV = \frac{C}{(1+r)} + \frac{C}{(1+r)^2} + \frac{C}{(1+r)^3} + \cdots

This is an infinite geometric series. The first term is $\frac{C}{1+r}$ , and the common ratio is $\frac{1}{1+r}$ . For an infinite geometric series where $|r| < 1$ , the sum is:

\text{Sum} = \frac{\text{first term}}{1 - \text{common ratio}}

So:

PV = \frac{\frac{C}{1+r}}{1 - \frac{1}{1+r}} = \frac{\frac{C}{1+r}}{\frac{r}{1+r}} = \frac{C}{r}

›Proof

Derivation using algebra:

Let $PV = \frac{C}{1+r} + \frac{C}{(1+r)^2} + \frac{C}{(1+r)^3} + \cdots$

Multiply both sides by $(1+r)$ :

$PV(1+r) = C + \frac{C}{1+r} + \frac{C}{(1+r)^2} + \cdots$

Notice the right side is $C + PV$ (since $PV$ is the same infinite sum starting from $\frac{C}{1+r}$ ).

So: $PV(1+r) = C + PV$

$PV + PV \cdot r = C + PV$

$PV \cdot r = C$

$PV = \frac{C}{r}$

The Critical Assumption

The formula $PV = C/r$ assumes the first payment occurs one period from today (an "ordinary perpetuity"). If the first payment occurs today (a "perpetuity due"), the formula becomes:

PV = C + \frac{C}{r}

Because you get the first payment immediately, and then the standard perpetuity starts from next period.

Watch out

A common mistake: students forget that the formula $PV = C/r$ assumes payments start at the end of the first period, not today. If a problem says "payments start now," you must add the immediate payment.

Real-World Examples …

Start your 14-day free trial to unlock the full solution →

Using the future-value-of-annuity formula with $i=0.03,\,n=16$ , the half-yearly deposit is about ₹2,480.57.

FV=R\cdot\dfrac{(1+i)^{n}-1}{i}

Semi-annual rate $i=\dfrac{6\%}{2}=0.03$ ; number of periods $n=8\times2=16$ .
$(1+i)^{16}=(1.03)^{16}=1.6047$ (given). …

Start your 14-day free trial to unlock the full solution →

Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CBSE 2025Set 465/W1XZY/41 markMCQ
$Q. The present value of a sequence of payments of ₹ 100 made at the end of every year and continuing forever, if the money is worth 5% compounded annually, is (A) ₹ 2,000 (B) ₹ 20,000 (C) ₹ 5,000 (D) ₹ 12,000$
›Reveal solutionSolution
Perpetuity present value $=\dfrac{R}{i}=\dfrac{100}{0.05}=\text{₹}2{,}000$ .

$PV=\dfrac{R}{i}$

Identify the data: payment $R=\text{₹}100$ at the end of each year, forever; rate $i=5\%=0.05$ . …
Start your 14-day free trial to see the full solution →
CBSE 2024Set 465/RQPS/41 markMCQ
$(1.03)^{16} = 1.6047$
›Reveal solutionSolution
Using the future-value-of-annuity formula with $i=0.03,\,n=16$ , the half-yearly deposit is about ₹2,480.57.

$FV=R\cdot\dfrac{(1+i)^{n}-1}{i}$

Semi-annual rate $i=\dfrac{6\%}{2}=0.03$ ; number of periods $n=8\times2=16$ .

$(1+i)^{16}=(1.03)^{16}=1.6047$ (given). …
Start your 14-day free trial to see the full solution →
CBSE 2024Set 465/S/RQPS/41 markMCQ
$Q. At what rate of interest will the present value of a perpetuity of ₹ 500 payable at the end of each quarter be ₹ 40,000 ? (A) 1.25% p.a. (B) 2.5% p.a. (C) 5% p.a. (D) 6% p.a.$
›Reveal solutionSolution
$PV=R/i$ gives a quarterly rate of 1.25%, so the annual rate is $4\times1.25\%=5\%$ .

$PV = \dfrac{R}{i}$

The payment is quarterly, so $i$ is the rate per quarter: $40000 = \dfrac{500}{i}$ . …
Start your 14-day free trial to see the full solution →
CBSE 2023Set 465/EF1GH/41 markMCQ
$i$
(a) $Ri$
(b) $R + \dfrac{R}{i}$
(c) $\dfrac{R}{i}$
(d) $R - Ri$
›Reveal solutionSolution
The present value of an ordinary (end-of-period) perpetuity of ₹R at rate $i$ is $\dfrac{R}{i}$ .

$PV=\dfrac{R}{(1+i)}+\dfrac{R}{(1+i)^2}+\cdots$

Sum the infinite geometric series: $PV=\dfrac{R/(1+i)}{1-\tfrac{1}{1+i}}$ . …
Start your 14-day free trial to see the full solution →

Q.What sum of money should be deposited at the end of every 6 months to accumulate ₹ 50,000 in 8 years, if money is worth 6% p.a. compounded semi-annually ? [Given : (1.03)16=1.6047(1.03)^{16} = 1.6047(1.03)16=1.6047] (A) ₹ 3,432.53 (B) ₹ 2,783.08 (C) ₹ 2,480.57 (D) ₹ 2,149.93

🎓Unlock everything free for 14 days

Q.What sum of money should be deposited at the end of every 6 months to accumulate ₹ 50,000 in 8 years, if money is worth 6% p.a. compounded semi-annually ? [Given : $(1.03)^{16} = 1.6047$ ] (A) ₹ 3,432.53 (B) ₹ 2,783.08 (C) ₹ 2,480.57 (D) ₹ 2,149.93

Unlock everything free for 14 days