find the numberterm in an Ap Given it first term and Last tem arek i3 and 23 respectively and that it commion difference is \( -2 \frac{1}{4} \)

To find the number of terms in an arithmetic progression (AP) given the first term, last term, and common difference, we can use the formula for the nth term of an AP: \[ a_n = a_1 + (n-1)d \] where: - \( a_n \) is the nth term, - \( a_1 \) is the first term, - \( d \) is the common difference, - \( n \) is the number of terms. Given: - First term, \( a_1 = 3 \), - Last term, \( a_n = 23 \), - Common difference, \( d = -2 \frac{1}{4} = -2.25 \). We can substitute these values into the formula and solve for \( n \) to find the number of terms in the AP. Solve the equation by following steps: - step0: Solve for \(n\): \(3+\left(n-1\right)\left(-2.25\right)=23\) - step1: Multiply the terms: \(3-2.25\left(n-1\right)=23\) - step2: Move the expression to the left side: \(3-2.25\left(n-1\right)-23=0\) - step3: Subtract the numbers: \(-20-2.25\left(n-1\right)=0\) - step4: Calculate: \(-17.75-2.25n=0\) - step5: Move the constant to the right side: \(-2.25n=0+17.75\) - step6: Add the terms: \(-2.25n=17.75\) - step7: Change the signs: \(2.25n=-17.75\) - step8: Divide both sides: \(\frac{2.25n}{2.25}=\frac{-17.75}{2.25}\) - step9: Divide the numbers: \(n=-\frac{71}{9}\) The number of terms in the arithmetic progression is \( n = -\frac{71}{9} \) or approximately \( n \approx -7.8 \). Since the number of terms cannot be negative or a fraction, we can conclude that there is no valid solution for the number of terms in this arithmetic progression.