In an A.P. the p th term is q and the (p+q)th term is 0 . Then the q th term is
(A) −p
(B p
(C) p+q
(D) p-q
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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 n -th term of an A.P. is given by the formula an=a+(n−1)d.
Step 2: Formulate equations from given conditions
According to the problem statement, the p -th term is q, so we can write a+(p−1)d=q. Also, the (p+q) -th term is 0, which gives us the equation a+(p+q−1)d=0. These are our two main equations.
Step 3: Solve for the common difference 'd'
To find the common difference d, we subtract equation (1) from equation (2). This eliminates a and allows us to solve for d. After simplifying, we find that d=−1.
Step 4: Solve for the first term 'a'
Now that we have the value of d, we can substitute it back into equation (1) to find the first term a. Substituting d=−1 into a+(p−1)d=q, we get a−p+1=q, which simplifies to a=p+q−1.
Step 5: Calculate the q-th term
Finally, we need to find the q -th term, aq. Using the formula aq=a+(q−1)d and substituting the values of a and d we found, we get aq=(p+q−1)+(q−1)(−1). This simplifies to p+q-1-q+1, which results in p.