The pth term of an A.P. is a and qth term is b. Prove that the sum of its (p+q) terms is 2p+q[a+b+p−qa−b]. $
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Step-by-Step Solution
Step 1: Define terms of an A.P.
Let A be the first term and D be the common difference of the Arithmetic Progression (A.P.). The nth term of an A.P. is given by the formula tn=A+(n−1)D.
Step 2: Formulate equations from given information
According to the problem statement, the pth term is a and the qth term is b. We can write these as two equations using the formula for the nth term.
Step 3: Solve for common difference D
Subtract equation (2) from equation (1) to eliminate A. This allows us to solve for the common difference D in terms of a, b, p, and q.
Step 4: Solve for the first term A
Substitute the value of D back into equation (1) to find the first term A. This expresses A in terms of a, b, p, and q.
Step 5: Apply sum formula for (p+q) terms
The sum of the first n terms of an A.P. is given by Sn=2n[2A+(n−1)D]. We need to find the sum of (p+q) terms, so we substitute n=p+q into this formula.
Step 6: Substitute A and D and simplify
Substitute the expressions for A and D into the sum formula. Then, simplify the terms inside the square brackets. Combine the terms involving p−qa−b and rearrange to match the required form.