Answer :
The common difference (d) of an arithmetic progression (AP), given the first term (a) and the last term (l), and the sum of all terms (s), is [tex]\( d = \frac{l - a}{n - 1} \),[/tex] where ( n ) is the number of terms in the AP.
1. Begin with the formula for the sum of an arithmetic progression (AP): [tex]\( s = \frac{n}{2}(a + l) \),[/tex] where ( s ) is the sum, ( n ) is the number of terms, ( a ) is the first term, and ( l ) is the last term.
2. Rearrange the formula to solve for ( n ): [tex]\( n = \frac{2s}{a + l} \).[/tex] This gives us the total number of terms in the AP.
3. The common difference ( d ) of an AP is given by the formula: [tex]\( d = \frac{l - a}{n - 1} \),[/tex] where ( n - 1 ) represents the number of intervals between terms.
4. Substitute the expression for ( n ) obtained in step 2 into the formula for ( d ): [tex]\( d = \frac{l - a}{\frac{2s}{a + l} - 1} \).[/tex] This expression represents the difference between consecutive terms in the AP.
5. Simplify the expression to find ( d ): [tex]\( d = \frac{l - a}{\frac{2s - (a + l)}{a + l}} \).[/tex] Here, we combine fractions to obtain a single fraction.
6. Further simplify the expression: [tex]\( d = \frac{(l - a)(a + l)}{2s - (a + l)} \).[/tex] This gives us the final formula for calculating the common difference ( d ).
Therefore, the common difference ( d ) can be calculated using this final formula, given the values of ( a ), ( l ), and ( s ). Each step in the derivation helps us understand how to derive the formula for the common difference of an arithmetic progression.
