In an AP it is given that


In an AP it is given that $S_{n}=q n^{2}$ and $S_{m}=q m^{2}$. Prove that $S_{q}=q^{3}$.


Given: $S_{n}=q n^{2}, S_{m}=q m^{2}$

To prove: $\mathrm{S}_{\mathrm{q}}=\mathrm{q}^{3}$

Put $n=1$ we get

$a=q \ldots \ldots$ equation 1

Put n = 2

2a + d = 4q ……equation 2

Using equation 1 and 2 we get

d = 2q

So $\mathrm{S}_{\mathrm{q}}=\frac{\mathrm{q}}{2}(2 \mathrm{q}+(\mathrm{q}-1) \times 2 \mathrm{q})$


Hence proved.


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