If $$\mathrm{S}(x)=(1+x)+2(1+x)^2+3(1+x)^3+\cdots+60(1+x)^{60}, x \neq 0$$, and $$(60)^2 \mathrm{~S}(60)=\mathrm{a}(\mathrm{b})^{\mathrm{b}}+\mathrm{b}$$, where $$a, b \in N$$, then $$(a+b)$$ equal to _________.

Let the first term of a series be $$T_1=6$$ and its $$r^{\text {th }}$$ term $$T_r=3 T_{r-1}+6^r, r=2,3$$, ............ $$n$$. If the sum of the first $$n$$ terms of this series is $$\frac{1}{5}\left(n^2-12 n+39\right)\left(4 \cdot 6^n-5 \cdot 3^n+1\right)$$, then $$n$$ is equal to ___________.

If $$1+\frac{\sqrt{3}-\sqrt{2}}{2 \sqrt{3}}+\frac{5-2 \sqrt{6}}{18}+\frac{9 \sqrt{3}-11 \sqrt{2}}{36 \sqrt{3}}+\frac{49-20 \sqrt{6}}{180}+\ldots$$ upto $$\infty=2+\left(\sqrt{\frac{b}{a}}+1\right) \log _e\left(\frac{a}{b}\right)$$, where a and b are integers with $$\operatorname{gcd}(a, b)=1$$, then $$\mathrm{11 a+18 b}$$ is equal to __________.

Let $$a_1, a_2, a_3, \ldots$$ be in an arithmetic progression of positive terms.

Let $$A_k=a_1^2-a_2^2+a_3^2-a_4^2+\ldots+a_{2 k-1}^2-a_{2 k}^2$$.

If $$\mathrm{A}_3=-153, \mathrm{~A}_5=-435$$ and $$\mathrm{a}_1^2+\mathrm{a}_2^2+\mathrm{a}_3^2=66$$, then $$\mathrm{a}_{17}-\mathrm{A}_7$$ is equal to ________.