Sum $1 + 2 a + 3 a^{2} + 4 a^{3} + . . . .$ to $n$ terms.

\frac{1 + (a^{n})}{(a - 1)^{2}} - \frac{n a^{n}}{1 + a}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

\frac{1 - 2 (a^{n})}{(a - 1)^{2}} + \frac{n a^{n}}{1 - 2 a}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

\frac{1 - (a^{n})}{(a - 1)^{2}} - \frac{n a^{n}}{1 - a}

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

none of these

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is B $\frac{1 - (a^{n})}{(a - 1)^{2}} - \frac{n a^{n}}{1 - a}$
Let $S = 1 + 2 a + 3 a^{2} + 4 a^{3} +$ .......... $+ n a^{n - 1}$
Multiply both sides by $a$ , we get
$S a = 0 + a + 2 a^{2} + 3 a^{3}$ ............. $(n - 1) a^{n - 1} + n a^{n}$
Subtract both equations,
$S (1 - a) = 1 + a + a^{2} + a^{3}$ ............. $a^{n - 1} - n a^{n}$
Clearly above series is G.P
Common ratio $= a$
$S (1 - a) = \frac{1 (a^{n} - 1)}{a - 1} - n a^{n}$
$S = \frac{1 - (a^{n})}{(a - 1)^{2}} - \frac{n a^{n}}{1 - a}$

Suggest Corrections

Similar questions

lim x \to - 1 \frac{\sqrt[3]{7 - x} - 2}{(x + 1)} = - \frac{1}{12}

lim x \to a = \frac{x^{n} - a^{n}}{x - a} = n a^{n - 1}

lim x \to a \frac{x^{n} - a^{n}}{x - a} = n a^{n - 1}

n \in N; a > 0

\int_{\frac{1}{n}}^{\frac{a n - 1}{n}} \frac{\sqrt{x}}{\sqrt{a - x} + \sqrt{x}} d x

lim x \to a \frac{x^{n} - a^{n}}{x - a} = n a^{n - 1}

n \in N, a > 0

\int_{1 / n}^{(a n - 1) / n} \frac{\sqrt{x}}{\sqrt{a - x} + \sqrt{x}} d x

Join BYJU'S Learning Program