SELECT * FROM question_mgmt as q WHERE id=1004 AND status=1 SELECT id,question_no,question,chapter FROM question_mgmt as q WHERE courseId=2 AND subId=6 AND chapterId=80 and ex_no='2' AND status=1 ORDER BY CAST(question_no AS UNSIGNED)

Question: If the sum of n terms of an A.P. is (pn + qn2), where p and q are constants, find the common difference

Answer:

It is known that,

\begin{align}   S_n= \frac{n}{2}\left[2a + (n-1)d\right] \end{align}

According to the given condition,

\begin{align}    \frac{n}{2}\left[2a + (n-1)d\right] = pn + qn^2 \end{align}

\begin{align}   ⇒ \frac{n}{2}\left[2a + nd-d\right] = pn + qn^2 \end{align}

\begin{align}   ⇒ na + n^2\frac{d}{2} - n.\frac{d}{2}= pn + qn^2 \end{align}

Comparing the coefficients of n2 on both sides, we obtain

\begin{align}   \frac{d}{2} = q \end{align}

\begin{align}  \therefore d = 2q \end{align}

Thus, the common difference of the A.P. is 2q.

 


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