Ten > Sequence and Series
Asked by Basanta · 2 years ago

Ramjanam repays a loan of Rs 189000 on installment based system by paying certain amount in the first month and then increases the installment by Rs 400 every month so that it takes 30 months to clear the loan. Find the amount of first installment.

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Atith Adhikari Atith Adhikari · 2 years ago
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Solution

This question involves the idea of the sum of n terms of an arithmetic series. According to the question, Ramjanam pays the total loan in installments where each installment can be represented as a, a + 400, a + 800, a + 1200, …, up to 30 terms.

Given:
- Total loan amount \( \rm S_{30} = 189000 \)
- Increase in installment amount every month \( \rm d = 400 \)

Let's denote:
- The first installment amount as \( \rm a_1 \)
- Total number of months as \( \rm n = 30 \)

We can use the formula for the sum of the first \( \rm n \) terms of an arithmetic series:

\( \rm S_n = \frac{n}{2}(2a_1 + (n - 1)d) \)

Substitute the given values:

\( \rm 189000 = \frac{30}{2}(2a_1 + (30 - 1) \times 400) \)

\( \rm 189000 = 15(2a_1 + 29 \times 400) \)

\( \rm 12600 = 2a_1 + 11600 \)

\( \rm 2a_1 = 12600 - 11600 \)

\( \rm 2a_1 = 1000 \)

\( \rm a_1 = \frac{1000}{2} \)

\( \rm a_1 = 500 \)

So, the amount of the first installment is Rs 500.

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