1
JEE Main 2022 (Online) 28th July Evening Shift
Numerical
+4
-1
Out of Syllabus
$${6 \over {{3^{12}}}} + {{10} \over {{3^{11}}}} + {{20} \over {{3^{10}}}} + {{40} \over {{3^9}}} + \,\,...\,\, + \,\,{{10240} \over 3} = {2^n}\,.\,m$$, where m is odd, then m . n is equal to ____________.
2
JEE Main 2022 (Online) 27th July Evening Shift
Numerical
+4
-1
Out of Syllabus

$$\frac{2^{3}-1^{3}}{1 \times 7}+\frac{4^{3}-3^{3}+2^{3}-1^{3}}{2 \times 11}+\frac{6^{3}-5^{3}+4^{3}-3^{3}+2^{3}-1^{3}}{3 \times 15}+\cdots+ \frac{30^{3}-29^{3}+28^{3}-27^{3}+\ldots+2^{3}-1^{3}}{15 \times 63}$$ is equal to _____________.

3
JEE Main 2022 (Online) 26th July Evening Shift
Numerical
+4
-1
Out of Syllabus

If $$\sum\limits_{k=1}^{10} \frac{k}{k^{4}+k^{2}+1}=\frac{m}{n}$$, where m and n are co-prime, then $$m+n$$ is equal to _____________.

4
JEE Main 2022 (Online) 26th July Evening Shift
Numerical
+4
-1

Different A.P.'s are constructed with the first term 100, the last term 199, and integral common differences. The sum of the common differences of all such A.P.'s having at least 3 terms and at most 33 terms is ___________.