Question

Sum of the first p, q and r terms of an A.P. are a, b and c, respectively.

Prove that

Answer 1

Let a₁ and d be the first term and the common difference of the A.P. respectively.

According to the given information,

Subtracting (2) from (1), we obtain

$$\begin{gathered} \left(p-1\right)d-\left(q-1\right)d=\frac{2a}{p}-\frac{2b}{q} \\ \Rightarrow d\left(p-1-q+1\right)=\frac{2aq-2bp}{pq} \\ \Rightarrow d\left(p-q\right)=\frac{2aq-2bp}{pq} \\ \Rightarrow d=\frac{2\left(aq-bp\right)}{pq\left(p-q\right)}.......\left(4\right) \end{gathered}$$

Subtracting (3) from (2), we obtain

Equating both the values of d obtained in (4) and (5), we obtain

$$\begin{gathered} \frac{aq-bp}{pq\left(p-q\right)}=\frac{br-qc}{qr\left(q-r\right)} \\ \Rightarrow \frac{aq-bp}{p\left(p-q\right)}=\frac{br-qc}{r\left(q-r\right)} \\ \Rightarrow r\left(q-r\right)\left(aq-bp\right)=p\left(p-q\right)\left(br-qc\right) \\ \Rightarrow r\left(aq-bp\right)\left(q-r\right)=p\left(br-qc\right)\left(p-q\right) \\ \Rightarrow \left(aqr-bpr\right)\left(q-r\right)=\left(bpr-cpq\right)\left(p-q\right) \end{gathered}$$

Dividing both sides by pqr, we obtain

Thus, the given result is proved.