A farmer borrows Gh 1,000 and agrees to repay with a total interest of Ghф 140 in 12 monthly instalments, each instalment being less than the preceding instalment by Ghф 1 What should be his first instalment?

To solve this problem, we need to determine the first installment amount that the farmer will pay back. Let's break down the problem step by step. ### Known Conditions: 1. Total amount borrowed: \( P = 1000 \) Gh 2. Total interest to be repaid: \( I = 140 \) Gh 3. Total amount to be repaid: \( A = P + I = 1000 + 140 = 1140 \) Gh 4. Number of installments: \( n = 12 \) 5. Each installment is less than the preceding installment by \( d = 1 \) Gh. ### Step 1: Define the Installments Let the first installment be \( x \) Gh. The subsequent installments will then be: - 1st installment: \( x \) - 2nd installment: \( x - 1 \) - 3rd installment: \( x - 2 \) - ... - 12th installment: \( x - 11 \) ### Step 2: Sum of the Installments The total amount paid in installments can be expressed as the sum of an arithmetic series: \[ S = x + (x - 1) + (x - 2) + \ldots + (x - 11) \] This can be simplified as: \[ S = 12x - (0 + 1 + 2 + \ldots + 11) \] The sum of the first \( n \) integers is given by the formula: \[ \text{Sum} = \frac{n(n + 1)}{2} \] For \( n = 11 \): \[ \text{Sum} = \frac{11 \times 12}{2} = 66 \] Thus, the total amount paid in installments becomes: \[ S = 12x - 66 \] ### Step 3: Set Up the Equation We know that the total amount to be repaid is \( 1140 \) Gh, so we set up the equation: \[ 12x - 66 = 1140 \] ### Step 4: Solve for \( x \) Now, we can solve for \( x \): \[ 12x = 1140 + 66 \] \[ 12x = 1206 \] \[ x = \frac{1206}{12} \] \[ x = 100.5 \] ### Conclusion The first installment should be \( 100.5 \) Gh. Thus, the farmer's first installment is **Gh 100.5**.