The first term of an arithmetic progression Ap_is_ 8. The ratio of the 7th term to the 9th term is 5:8. Calculate the common difference of the progression (WAEC).

Answer 1

Let the first term of the arithmetic progression (AP) be denoted as \( a = 8 \) and let the common difference be denoted as \( d \).

The \( n \)-th term of an arithmetic progression can be expressed as: \[ A_n = a + (n-1)d \]

Now, we can write expressions for the 7th term and the 9th term:

1. The 7th term \( A_7 \): \[ A_7 = a + (7 - 1)d = 8 + 6d = 8 + 6d \]

2. The 9th term \( A_9 \): \[ A_9 = a + (9 - 1)d = 8 + 8d = 8 + 8d \]

According to the problem, the ratio of the 7th term to the 9th term is given as \( \frac{5}{8} \). Thus, we can set up the following equation: \[ \frac{A_7}{A_9} = \frac{5}{8} \]

Substituting \( A_7 \) and \( A_9 \): \[ \frac{8 + 6d}{8 + 8d} = \frac{5}{8} \]

To solve for \( d \), we can cross-multiply: \[ 8(8 + 6d) = 5(8 + 8d) \]

Expanding both sides: \[ 64 + 48d = 40 + 40d \]

Now, let's isolate \( d \) by moving the \( d \) terms to one side: \[ 48d - 40d = 40 - 64 \]

This simplifies to: \[ 8d = -24 \]

Dividing both sides by 8: \[ d = -3 \]

Thus, the common difference of the arithmetic progression is: \[ \boxed{-3} \]