Sequences and S...

The value of $$\displaystyle \sum_{r = 16}^{30}{ (r + 2)(r - 3) } $$ is equal to :

The sum of the following series $$1+6+\dfrac{9(1^2+2^2+3^2)}{7}+\dfrac{12(1^2+2^2+3^2+4^2)}{9}+\dfrac{15(1^2+2^2+....+5^2)}{11}+.....$$ up to $$15$$ terms is:

Let $$A$$ be the sum of the first $$20$$ terms and $$B$$ be the sum of the first $$40$$ terms of the series $$1^{2} + 2.2^{2} + 3^{2} + 2.4^{2} + 5^{2} + 2.6^{2} + .....$$ If $$B - 2A = 100\lambda$$, then $$\lambda$$ is equal to

If the sum of the first 15 terms of the series $$\lgroup \frac{3}{4} \rgroup^3 + \lgroup 1\frac{1}{2} \rgroup^3 + \lgroup 2\frac{1}{4} \rgroup^3 + 3^3 + \lgroup 3\frac{3}{4} \rgroup^3 + ...........$$ is equal to 225 k. then k is equal to:

Let  $$S _ { k } = \dfrac { 1 + 2 + 3 + \ldots + k } { k } .$$  If   $${ S }_{ { 1 } }^{ { 2 } }+{ S }_{ { 2 } }^{ { 2 } }++{ S }_{ { 10 } }^{ { 2 } }=\dfrac { 5 }{ 12 } { A }$$  then  $$A$$  is equal to :

The sum $$\mathrm{V}_{1}+\mathrm{V}_{2}+\ldots+\mathrm{V}_{\mathrm{n}}$$ is 

Let $$\mathrm{a}_{1},\ \mathrm{a}_{2},\ \mathrm{a}_{3},\ \ldots,\ \mathrm{a}_{11}$$ be real numbers satisfying $$\mathrm{a}_{1}=15,27-2\mathrm{a}_{2}>0$$ and $$\mathrm{a}_{\mathrm{k}}=2\mathrm{a}_{\mathrm{k}-1}-\mathrm{a}_{\mathrm{k}-2}$$ 

$$x\left (x + \dfrac {1}{x}\right )^{2} + \left (x^{2} + \dfrac {1}{x^{2}}\right )^{2} + \left (x^{3} + \dfrac {1}{x^{3}}\right )^{2}$$ .... upto $$n$$ terms is

Which letter will be the fifth from the right if the first and second, the third and fourth and so on are interchanged among each other in the word "COMPANIONATE"?

Mean of the first $$n$$ terms of the A.P. $$a, (a + d), (a + 2d), ........$$ is