Q. 223.9( 72 Votes )

# If the p^{th}, q^{th} and r^{th}terms of a G.P are a, b and c, respectively. Prove that

a^{q – r} × b^{r – p} × c^{P – q} = 1.

Answer :

Given p^{th}, q^{th} and r^{th}terms of a G.P are a, b and c, respectively

Here

a_{p} = a = ar^{p-1}

a_{q} = b = ar^{q-1}

a_{r} = c = ar^{r-1}

Now,

a^{q – r} × b^{r – p} × c^{P – q} = (ar^{p - 1})^{q - r} × (ar^{q - 1})^{r - p} × (ar^{r-1})^{p – q}

⇒ a^{q – r} × b^{r – p} × c^{P – q} = (a^{(q - r)} × r^{(p – 1)(q-r)}) × (a^{(r - p)} × r^{(q – 1)(r - p)}) × (a^{(p - q)} × r^{(r – 1)(p - q)})

⇒ a^{q – r} × b^{r – p} × c^{P – q} = (a^{(q – r)} × r^{(pq – q - pr +r)}) × (a^{(r - p)} × r^{(qr – r - pq + p )}) × (a^{(p - q)} × r^{(pr – p – qr + q)})

⇒ a^{q – r} × b^{r – p} × c^{P – q} = (a^{(q – r + r – p + p - q)} × r^{(pq – q – pr +r + qr - r –pq + p +pr – p –qr + q)})

⇒ a^{q – r} × b^{r – p} × c^{P – q} = (a^{0} × r^{0}) = 1

∴ a^{q – r} × b^{r – p} × c^{P – q} = 1

Hence proved.

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