Sequences and S...

If $${ b }_{ i }=1-{ a }_{ i },na=\sum _{ i=1 }^{ n }{ { a }_{ i } } ,nb=\sum _{ i=1 }^{ n }{ { b }_{ i } } \quad $$, then $$\sum _{ i=1 }^{ n }{ { { a }_{ i }b }_{ i } } +\sum _{ i=1 }^{ n }{ { \left( { a }_{ i }-a \right)  }^{ 2 } } =$$

The sum of the first $$n$$ terms of the series $${ 1 }^{ 2 }+2\cdot { 2 }^{ 2 }+{ 3 }^{ 3 }+2\cdot { 4 }^{ 2 }+{ 5 }^{ 2 }+2\cdot { 6 }^{ 2 }+\cdots $$ is $$\dfrac { n{ \left( n+1 \right)  }^{ 2 } }{ 2 } $$ when $$n$$ is even, when $$n$$ is odd the sum is

$$4,9,25,?,121,169$$

Find the missing number in the circle:

If $$\displaystyle \sum_{r = 1}^{n}t_{n} = \dfrac {n(n +1)(n + 2)(n + 3)}{8}$$, then $$\displaystyle \sum_{r = 1}^{n} \dfrac {1}{t_{1}}$$ equals

Select the INCORRECT match

Let $$r^{th} $$ term of a series is given by, $$T_r = \dfrac {r}{1-3r^2 + r^4} .$$

Figures $$1$$ and $$2$$ are related in a particular manner. Establish the same relationship between figures $$3$$ and $$4$$ by choosing a figure from amongst the options.

The value of $$\sum _{ n=1 }^{ \infty  }{ { \left( -1 \right)  }^{ n+1 }\left( \cfrac { n }{ { 5 }^{ n } }  \right)  } $$ equals

Let S be the infinite sum given by $$S=\displaystyle \sum_{n=0}^{\infty}\frac{a_n}{10^{2n}}$$, where $$(a_n)_{n\geq 0}$$ is a sequence defined by $$a_0=a_1=1$$ and $$a_j=20a_{j-1}$$ for $$j\geq 2$$. If $$S$$ is expressed in the form $$\displaystyle\frac{a}{b}$$, where $$a, b$$ are coprime positive integers, than $$a$$ equals.