1
JEE Main 2023 (Online) 11th April Evening Shift
Numerical
+4
-1 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 _________.

2
JEE Main 2023 (Online) 11th April Morning Shift
Numerical
+4
-1 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 ___________.

3
JEE Main 2023 (Online) 10th April Evening Shift
Numerical
+4
-1 Suppose $$a_{1}, a_{2}, 2, a_{3}, a_{4}$$ be in an arithmetico-geometric progression. If the common ratio of the corresponding geometric progression is 2 and the sum of all 5 terms of the arithmetico-geometric progression is $$\frac{49}{2}$$, then $$a_{4}$$ is equal to __________.

4
JEE Main 2023 (Online) 10th April Morning Shift
Numerical
+4
-1 The sum of all those terms, of the arithmetic progression 3, 8, 13, ...., 373, which are not divisible by 3, is equal to ____________.

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