If the sum of the first m terms of an AP is the same as the sum of its first n terms (m=n), show that the sum of its first (m + n) terms is: Sm+n=0
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Step-by-Step Solution
Step 1: Write the formula for the sum of an AP
The sum of the first k terms of an arithmetic progression (AP) is given by the formula Sk=2k[2a+(k−1)d], where a is the first term and d is the common difference.
Step 2: Set up the given condition
We are given that the sum of the first m terms is equal to the sum of the first n terms. We substitute k=m and k=n into the sum formula and set them equal.
Step 3: Simplify the equation
First, multiply both sides by 2 to clear the denominators. Then, rearrange the terms to group 2a and d. Factor out common terms, noting that m2−n2=(m−n)(m+n). Since m=n, we know that m−n=0, allowing us to divide by (m-n).
Step 4: Derive the relationship between 'a' and 'd'
Since m=n, we can divide both sides of the equation by (m-n). This gives us a crucial relationship between the first term a, the common difference d, and the numbers of terms m and n.
Step 5: Calculate the sum of the first (m+n) terms
Now, we need to find the sum of the first (m+n) terms. We use the sum formula again, substituting k=(m+n).
Step 6: Substitute the derived relationship
From the previous step, we found that 2a+(m+n−1)d=0. We substitute this into the expression for Sm+n. Multiplying by zero results in zero.