Question

If $3000 is deposited at the end of each half year in an account that earns 6.2% compounded semiannually, after how many half years will the account contain $170,000? (round your answer up to the nearest half year.)

Answer 1

To calculate the number of half years needed for the account to reach $170,000, we can use the compound interest formula.

Step-by-step explanation:

To determine the number of half years required for the account to reach $170,000, we can use the compound interest formula: A = P(1 + r/n)^(nt), where A is the final amount, P is the principal, r is the interest rate, n is the number of times interest is compounded per year, and t is the time in years.

In this case, the principal is $3000, the interest rate is 6.2%, and interest is compounded semiannually, so n is 2.

We need to solve for t.

Starting with the formula: $170,000 = $3000(1 + 0.062/2)^(2t), we can simplify it to: 56.67 = (1.031)^2t.

To isolate t, we take the logarithm of both sides: log(56.67) = log[(1.031)^2t].

Using logarithm properties, we can bring down the exponent: log(56.67) = 2t * log(1.031).

Dividing both sides by 2*log(1.031), we get: t = log(56.67) / (2 * log(1.031)).

Using a calculator, we find t ≈ 25.83.

Rounding up to the nearest half year, it will take approximately 26 half years for the account to contain $170,000.