find the number term in an Ap Given it finst term and Last term are 'is and 23 respectively and that it common difference is \( -\frac{1}{4} \)

To find the number of terms in an arithmetic progression (AP), we can use the formula for the \( n \)-th term of an AP: \[ a_n = a + (n-1)d \] where: - \( a_n \) is the \( n \)-th term, - \( a \) is the first term, - \( d \) is the common difference, - \( n \) is the number of terms. In your case: - The first term \( a = 15 \), - The last term \( a_n = 23 \), - The common difference \( d = -\frac{1}{4} \). We need to find \( n \) such that: \[ 23 = 15 + (n-1)\left(-\frac{1}{4}\right) \] Now, let's solve for \( n \): 1. Subtract 15 from both sides: \[ 23 - 15 = (n-1)\left(-\frac{1}{4}\right) \] \[ 8 = (n-1)\left(-\frac{1}{4}\right) \] 2. Multiply both sides by -4 to eliminate the fraction: \[ 8 \times -4 = n - 1 \] \[ -32 = n - 1 \] 3. Add 1 to both sides: \[ -32 + 1 = n \] \[ n = -31 \] Since the number of terms cannot be negative, it seems there is an inconsistency in the values provided. Please double-check the values of the first term, last term, and common difference to ensure they are correct. If they are, then the arithmetic progression as described does not yield a valid number of terms.