Two A.P.s have the same first term. The common difference of the first A.P. is -3 and of the second A.P. is -5. The difference of the $6^{th}$ term of the second A.P. from that of the first A.P. is :

Step 1: Define the terms of an A.P.

We start by recalling the formula for the $n^{th}$ term of an arithmetic progression (A.P.), where $a_n$ is the $n^{th}$ term, $a$ is the first term, and $d$ is the common difference.

Step 2: Express the 6th terms for both A.P.s

Let $a$ be the common first term for both A.P.s. For the first A.P., the common difference is $d_1 = -3$. For the second A.P., the common difference is $d_2 = -5$. We substitute these values into the formula to find the $6^{th}$ term for each A.P.

Step 3: Calculate the difference between the 6th terms

The problem asks for the difference of the $6^{th}$ term of the second A.P. from that of the first A.P., which means we need to calculate $a_{1{,}6} - a_{2{,}6}$. We substitute the expressions we found for each $6^{th}$ term.

Step 4: Simplify the expression

We simplify the expression by distributing the negative sign and combining like terms. The first term $a$ cancels out, leaving us with the final difference.

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