1
JEE Main 2023 (Online) 15th April Morning Shift
Numerical
+4
-1
Out of Syllabus
If the sum of the series

$\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{2^{2}}-\frac{1}{2 \cdot 3}+\frac{1}{3^{2}}\right)+\left(\frac{1}{2^{3}}-\frac{1}{2^{2} \cdot 3}+\frac{1}{2 \cdot 3^{2}}-\frac{1}{3^{3}}\right)+$

$\left(\frac{1}{2^{4}}-\frac{1}{2^{3} \cdot 3}+\frac{1}{2^{2} \cdot 3^{2}}-\frac{1}{2 \cdot 3^{3}}+\frac{1}{3^{4}}\right)+\ldots$

is $\frac{\alpha}{\beta}$, where $\alpha$ and $\beta$ are co-prime, then $\alpha+3 \beta$ is equal to __________.
2
JEE Main 2023 (Online) 13th April Morning Shift
Numerical
+4
-1
Out of Syllabus

The sum to $$20$$ terms of the series $$2 \cdot 2^{2}-3^{2}+2 \cdot 4^{2}-5^{2}+2 \cdot 6^{2}-\ldots \ldots$$ is equal to __________.

3
JEE Main 2023 (Online) 11th April Evening Shift
Numerical
+4
-1
Out of Syllabus

For $$k \in \mathbb{N}$$, if the sum of the series $$1+\frac{4}{k}+\frac{8}{k^{2}}+\frac{13}{k^{3}}+\frac{19}{k^{4}}+\ldots$$ is 10 , then the value of $$k$$ is _________.

4
JEE Main 2023 (Online) 11th April Morning Shift
Numerical
+4
-1
Out of Syllabus

Let $$S=109+\frac{108}{5}+\frac{107}{5^{2}}+\ldots .+\frac{2}{5^{107}}+\frac{1}{5^{108}}$$. Then the value of $$\left(16 S-(25)^{-54}\right)$$ is equal to ___________.