Find the sum n=0r qn=1+q+q2++qr.

$\frac1{q-1}+\frac{q^{r+1}}{q-1}$

GATE 2025 AG – Question 36

Engineering Mathematics · Calculus · 2 marks · Multiple choice

Find the sum $\sum_{n=0}^r q^n=1+q+q^2+\cdots+q^r$.

$\frac1{1-q}-\frac{q^{r+1}}{1-q}$
$\frac1{q-1}+\frac{q^{r+1}}{q-1}$
$\frac1{1-q}+\frac{q^{r+1}}{1-q}$
$\frac1{q-1}$

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Correct answer: (A) $\frac1{1-q}-\frac{q^{r+1}}{1-q}$

Explanation

Let the finite sum be S. Multiplying by q and subtracting cancels intermediate terms: $(1-q)S=1-q^{r+1}$. Hence $S=(1-q^{r+1})/(1-q)$ for $q\ne1$,which is **A**. For the special case q=1,the sum is r+1; the displayed rational forms are then understood by a limit.

GATE 2025 AG – Question 36

Explanation

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