If p^ th term of an AP is k , then what is the sum of p^ th term, (p+q)^ th term and (p-q)^ th term of the AP ?

Step-by-step solution

Let the first term of the AP be a and the common difference be d . The p^ th term is given by T_p = a + (p-1)d = k . The (p+q)^ th term is T_ p+q = a + (p+q-1)d . The (p-q)^ th term is T_ p-q = a + (p-q-1)d . The required sum is T_p + T_ p+q + T_ p-q . Substituting the values, we get: [a + (p-1)d] + [a + (p+q-1)d] + [a + (p-q-1)d] = 3a + d[(p-1) + (p+q-1) + (p-q-1)] = 3a + d(3p - 3) = 3[a + (p-1)d] Since a + (p-1)d = k , the sum is 3k . Answer: 3k