Ajesh has set up a sinking fund in order to have ₹ 10,00,000 after 10 years for his son's education. The amount should be set aside at the end of every 6 months into an account paying 5% per annum compounded half yearly is: [given $(1.025)^{20} = 1.6386$]

Ajesh wants to save ₹ 10,00,000 for his son's education after 10 years by depositing equal amounts every 6 months into an account with 5% per annum interest compounded half-yearly.

Given information:

• Target Amount (Future Value) = ₹ 10,00,000
• Time Period = 10 years
• Deposits made every 6 months
• Interest Rate = 5% per annum compounded half-yearly
• $(1.025)^{20} = 1.6386$

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Converting to half-yearly terms:

Number of deposits: $n = 10 \times 2 = 20$

Interest rate per 6 months: $i = \frac{5\%}{2} = 2.5\% = 0.025$

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The sinking fund formula relates the future value to regular deposits:

$\text{Future Value} = P \times \frac{(1+i)^n - 1}{i}$

where $P$ is the amount to deposit every 6 months.

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Substituting the values:

$10,00,000 = P \times \frac{(1.025)^{20} - 1}{0.025}$

$10,00,000 = P \times \frac{1.6386 - 1}{0.025}$

$10,00,000 = P \times \frac{0.6386}{0.025}$

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Simplifying:

$\frac{0.6386}{0.025} = \frac{638.6}{25} = 25.544$

$10,00,000 = P \times 25.544$

$P = \frac{10,00,000}{25.544}$

$P = 39,148.1$

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Therefore, Ajesh needs to deposit ₹ 39,148.1 every 6 months.

This matches Option 3: ₹ 3,9,148.1 (in Indian numbering format).