If there are (2n+1) terms in an A.P., then prove that the ratio of the sum of odd terms and the sum of even terms is (n+1): n
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Step-by-Step Solution
Step 1: Define the A.P. and its terms
Let the arithmetic progression (A.P.) have (2n+1) terms. We can denote these terms as a1,a2,…,a2n+1. Let a be the first term and d be the common difference of this A.P.
Step 2: Identify odd and even terms
The odd terms are a1,a3,…,a2n+1. There are (n+1) odd terms. The even terms are a2,a4,…,a2n. There are n even terms.
Step 3: Calculate the sum of odd terms
The odd terms form an A.P. with first term a1=a and common difference 2d. There are (n+1) such terms. Using the sum formula for an A.P., Sk=2k[2A+(k−1)D], where A is the first term and D is the common difference, we find the sum of odd terms.
Step 4: Calculate the sum of even terms
The even terms form an A.P. with first term a2=a+d and common difference 2d. There are n such terms. Applying the sum formula for an A.P., we find the sum of even terms.
Step 5: Find the ratio of the sums
Finally, we find the ratio of the sum of odd terms to the sum of even terms by dividing the expressions we derived. The common factor (a+nd) cancels out, leaving us with the desired ratio.