1
JEE Main 2024 (Online) 30th January Evening Shift
Numerical
+4
-1

Let $$S_n$$ be the sum to $$n$$-terms of an arithmetic progression $$3,7,11$$, If $$40<\left(\frac{6}{n(n+1)} \sum_\limits{k=1}^n S_k\right)<42$$, then $$n$$ equals ________.

2
JEE Main 2024 (Online) 30th January Morning Shift
Numerical
+4
-1

Let $$\alpha=1^2+4^2+8^2+13^2+19^2+26^2+\ldots$$ upto 10 terms and $$\beta=\sum_\limits{n=1}^{10} n^4$$. If $$4 \alpha-\beta=55 k+40$$, then $$\mathrm{k}$$ is equal to __________.

3
JEE Main 2024 (Online) 27th January Morning Shift
Numerical
+4
-1
If $8=3+\frac{1}{4}(3+p)+\frac{1}{4^2}(3+2 p)+\frac{1}{4^3}(3+3 p)+\cdots \cdots \infty$, then the value of $p$ is ____________.
4
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 __________.