For an arithmetic sequence $\left\{ a _ { n } \right\}$ with first term 0 and common difference not equal to 0, a sequence $\left\{ b _ { n } \right\}$ satisfies $a _ { n + 1 } b _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$. Find the value of $b _ { 27 }$. [4 points]

For an arithmetic sequence $\left\{ a _ { n } \right\}$ with first term 0 and nonzero common difference, the sequence $\left\{ b _ { n } \right\}$ satisfies $a _ { n + 1 } b _ { n } = \sum _ { k = 1 } ^ { n } a _ { k }$. Find the value of $b _ { 27 }$. [4 points]

There is a rectangle $\mathrm { A } _ { 1 } \mathrm {~B} _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$ with $\overline { \mathrm { A } _ { 1 } \mathrm {~B} _ { 1 } } = 1$ and $\overline { \mathrm {~B} _ { 1 } \mathrm { C } _ { 1 } } = 2$. As shown in the figure, let $\mathrm { M } _ { 1 }$ be the midpoint of segment $\mathrm { B } _ { 1 } \mathrm { C } _ { 1 }$, and on segment $\mathrm { A } _ { 1 } \mathrm { D } _ { 1 }$, two points $\mathrm { B } _ { 2 } , \mathrm { C } _ { 2 }$ are determined such that $\angle \mathrm { A } _ { 1 } \mathrm { M } _ { 1 } \mathrm {~B} _ { 2 } = \angle \mathrm { C } _ { 2 } \mathrm { M } _ { 1 } \mathrm { D } _ { 1 } = 15 ^ { \circ } , \angle \mathrm { B } _ { 2 } \mathrm { M } _ { 1 } \mathrm { C } _ { 2 } = 60 ^ { \circ }$. Let $S _ { 1 }$ be the sum of the area of triangle $\mathrm { A } _ { 1 } \mathrm { M } _ { 1 } \mathrm {~B} _ { 2 }$ and the area of triangle $\mathrm { C } _ { 2 } \mathrm { M } _ { 1 } \mathrm { D } _ { 1 }$.
Quadrilateral $\mathrm { A } _ { 2 } \mathrm {~B} _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ is a rectangle with $\overline { \mathrm { B } _ { 2 } \mathrm { C } _ { 2 } } = 2 \overline { \mathrm {~A} _ { 2 } \mathrm {~B} _ { 2 } }$ such that two points $\mathrm { A } _ { 2 } , \mathrm { D } _ { 2 }$ are determined as shown in the figure. Let $\mathrm { M } _ { 2 }$ be the midpoint of segment $\mathrm { B } _ { 2 } \mathrm { C } _ { 2 }$, and on segment $\mathrm { A } _ { 2 } \mathrm { D } _ { 2 }$, two points $\mathrm { B } _ { 3 } , \mathrm { C } _ { 3 }$ are determined such that $\angle \mathrm { A } _ { 2 } \mathrm { M } _ { 2 } \mathrm {~B} _ { 3 } = \angle \mathrm { C } _ { 3 } \mathrm { M } _ { 2 } \mathrm { D } _ { 2 } = 15 ^ { \circ }$, $\angle \mathrm { B } _ { 3 } \mathrm { M } _ { 2 } \mathrm { C } _ { 3 } = 60 ^ { \circ }$. Let $S _ { 2 }$ be the sum of the area of triangle $\mathrm { A } _ { 2 } \mathrm { M } _ { 2 } \mathrm {~B} _ { 3 }$ and the area of triangle $\mathrm { C } _ { 3 } \mathrm { M } _ { 2 } \mathrm { D } _ { 2 }$. Continuing this process, what is the value of $\sum _ { n = 1 } ^ { \infty } S _ { n }$ for the obtained $S _ { n }$? [4 points]
(1) $\frac { 2 + \sqrt { 3 } } { 6 }$
(2) $\frac { 3 - \sqrt { 3 } } { 2 }$
(3) $\frac { 4 + \sqrt { 3 } } { 9 }$
(4) $\frac { 5 - \sqrt { 3 } } { 5 }$
(5) $\frac { 7 - \sqrt { 3 } } { 8 }$

For an arithmetic sequence $\left\{ a _ { n } \right\}$ with first term 3, if $\sum _ { k = 1 } ^ { 5 } a _ { k } = 55$, find the value of $\sum _ { k = 1 } ^ { 5 } k \left( a _ { k } - 3 \right)$. [3 points]

A sequence $\left\{ a_{n} \right\}$ with $a_{1} = 2$ and an arithmetic sequence $\left\{ b_{n} \right\}$ with $b_{1} = 2$ satisfy $$\sum_{k=1}^{n} \frac{a_{k}}{b_{k+1}} = \frac{1}{2}n^{2}$$ for all natural numbers $n$. What is the value of $\sum_{k=1}^{5} a_{k}$? [4 points]
(1) 120
(2) 125
(3) 130
(4) 135
(5) 140

If the $n$ terms $a _ { 1 } , a _ { 2 } , \ldots , a _ { n }$ are in arithmetic progression with increment $r$, then the difference between the mean of their squares and the square of their mean is
(A) $\frac { r ^ { 2 } \left( ( n - 1 ) ^ { 2 } - 1 \right) } { 12 }$
(B) $\frac { r ^ { 2 } } { 12 }$
(C) $\frac { r ^ { 2 } \left( n ^ { 2 } - 1 \right) } { 12 }$
(D) $\frac { n ^ { 2 } - 1 } { 12 }$

If the sum of first $n$ terms of an A.P. is $cn^{2}$, then the sum of squares of these $n$ terms is
(A) $\frac{n\left(4n^{2}-1\right)c^{2}}{6}$
(B) $\frac{n\left(4n^{2}+1\right)c^{2}}{3}$
(C) $\frac{n\left(4n^{2}-1\right)c^{2}}{3}$
(D) $\frac{n\left(4n^{2}+1\right)c^{2}}{6}$

Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots , a _ { 11 }$ be real numbers satisfying $\mathrm { a } _ { 1 } = 15 , \quad 27 - 2 \mathrm { a } _ { 2 } > 0$ and $\mathrm { a } _ { \mathrm { k } } = 2 \mathrm { a } _ { \mathrm { k } - 1 } - \mathrm { a } _ { \mathrm { k } - 2 }$ for $\mathrm { k } = 3,4 , \ldots , 11$.
If $\frac { a _ { 1 } ^ { 2 } + a _ { 2 } ^ { 2 } + \ldots + a _ { 11 } ^ { 2 } } { 11 } = 90$, then the value of $\frac { a _ { 1 } + a _ { 2 } + \ldots + a _ { 11 } } { 11 }$ is equal to

Let $l _ { 1 } , l _ { 2 } , \ldots , l _ { 100 }$ be consecutive terms of an arithmetic progression with common difference $d _ { 1 }$, and let $w _ { 1 } , w _ { 2 } , \ldots , w _ { 100 }$ be consecutive terms of another arithmetic progression with common difference $d _ { 2 }$, where $d _ { 1 } d _ { 2 } = 10$. For each $i = 1,2 , \ldots , 100$, let $R _ { i }$ be a rectangle with length $l _ { i }$, width $w _ { i }$ and area $A _ { i }$. If $A _ { 51 } - A _ { 50 } = 1000$, then the value of $A _ { 100 } - A _ { 90 }$ is $\_\_\_\_$.

If $( 10 ) ^ { 9 } + 2 ( 11 ) ^ { 1 } ( 10 ) ^ { 8 } + 3 ( 11 ) ^ { 2 } ( 10 ) ^ { 7 } + \ldots\ldots + 10 ( 11 ) ^ { 9 } = k ( 10 ) ^ { 9 }$, then $k$ is equal to:
(1) 100
(2) 110
(3) $\frac { 121 } { 10 }$
(4) $\frac { 441 } { 100 }$

Let $a, b, c \in \mathbb{R}$. If $f(x) = ax^2 + bx + c$ is such that $a + b + c = 3$ and $f(x + y) = f(x) + f(y) + xy,\ \forall x, y \in \mathbb{R}$, then $\displaystyle\sum_{n=1}^{10} f(n)$ is equal to:
(1) 330
(2) 165
(3) 190
(4) 255

Let $A$ be the sum of the first 20 terms and $B$ be the sum of the first 40 terms of the series $1 ^ { 2 } + 2 \cdot 2 ^ { 2 } + 3 ^ { 2 } + 2 \cdot 4 ^ { 2 } + 5 ^ { 2 } + 2 \cdot 6 ^ { 2 } + \ldots$ If $B - 2 A = 100 \lambda$, then $\lambda$ is equal to :
(1) 496
(2) 232
(3) 248
(4) 464

Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots \ldots , a _ { 49 }$ be in $A.P$. such that $\sum _ { k = 0 } ^ { 12 } a _ { 4 k + 1 } = 416$ and $a _ { 9 } + a _ { 43 } = 66$. If $a _ { 1 } ^ { 2 } + a _ { 2 } ^ { 2 } + \ldots + a _ { 17 } ^ { 2 } = 140 m$, then $m$ is equal to:
(1) 33
(2) 66
(3) 68
(4) 34

If the sum of the first 15 terms of the series $\left( \frac { 3 } { 4 } \right) ^ { 3 } + \left( 1 \frac { 1 } { 2 } \right) ^ { 3 } + \left( 2 \frac { 1 } { 4 } \right) ^ { 3 } + 3 ^ { 3 } + \left( 3 \frac { 3 } { 4 } \right) ^ { 3 } + \ldots$ is equal to 225 K , then $K$ is equal to :
(1) 9
(2) 27
(3) 54
(4) 108

The sum $\sum _ { k = 1 } ^ { 20 } k \frac { 1 } { 2 ^ { k } }$ is equal to
(1) $1 - \frac { 11 } { 3 ^ { 20 } }$
(2) $2 - \frac { 21 } { 2 ^ { 20 } }$
(3) $2 - \frac { 3 } { 2 ^ { 17 } }$
(4) $2 - \frac { 11 } { 2 ^ { 19 } }$

The sum of the series $2\cdot{}^{20}C_0 + 5\cdot{}^{20}C_1 + 8\cdot{}^{20}C_2 + 11\cdot{}^{20}C_3 + \ldots + 62\cdot{}^{20}C_{20}$ is equal to
(1) $2^{26}$
(2) $2^{25}$
(3) $2^{24}$
(4) $2^{23}$

The sum of the series $1 + 2 \times 3 + 3 \times 5 + 4 \times 7 + \ldots$ upto $11 ^ { \text {th} }$ term is:
(1) 945
(2) 916
(3) 946
(4) 915

If $1 + \left( 1 - 2 ^ { 2 } \cdot 1 \right) + \left( 1 - 4 ^ { 2 } \cdot 3 \right) + \left( 1 - 6 ^ { 2 } \cdot 5 \right) + \ldots \ldots + \left( 1 - 20 ^ { 2 } \cdot 19 \right) = \alpha - 220 \beta$, then an ordered pair $( \alpha , \beta )$ is equal to:
(1) $( 10,97 )$
(2) $( 11,103 )$
(3) $( 10,103 )$
(4) $( 11,97 )$

If $\alpha , \beta$ are natural numbers such that $100 ^ { \alpha } - 199 \beta = ( 100 ) ( 100 ) + ( 99 ) ( 101 ) + ( 98 ) ( 102 ) + \ldots . + ( 1 ) ( 199 )$, then the slope of the line passing through $( \alpha , \beta )$ and origin is:
(1) 540
(2) 550
(3) 530
(4) 510

Let $a _ { 1 } = b _ { 1 } = 1$, $a _ { n } = a _ { n - 1 } + 2$ and $b _ { n } = a _ { n } + b _ { n - 1 }$ for every natural number $n \geq 2$. Then $\sum _ { n = 1 } ^ { 15 } a _ { n } \cdot b _ { n }$ is equal to $\_\_\_\_$.

Let $S _ { K } = \frac { 1 + 2 + \ldots + K } { K }$ and $\sum _ { j = 1 } ^ { n } S ^ { 2 } { } _ { j } = \frac { n } { A } \left( B n ^ { 2 } + C n + D \right)$ where $A , B , C , D \in N$ and $A$ has least value then
(1) $A + C + D$ is not divisible by $D$
(2) $A + B = 5 ( D - C )$
(3) $A + B + C + D$ is divisible by 5
(4) $A + B$ is divisible by $D$

$\frac { 6 } { 3 ^ { 26 } } + \frac { 10 } { 3 ^ { 25 } } + \frac { 10.2 } { 3 ^ { 24 } } + \frac { 10.2 ^ { 2 } } { 3 ^ { 23 } } + \cdots + \frac { 10.2 ^ { 24 } } { 3 }$ is equal to

Consider a sequence $\{a_n\}$ $(n = 1, 2, 3, \cdots)$ where the sum of the first $n$ terms is
$$\sum_{k=1}^{n} a_k = n^2 + 3n$$
(1) Then $a_n = \mathbf{A}\, n + \mathbf{B}$.
(2) For the sequence $\{b_n\}$ $(n = 1, 2, 3, \cdots)$, where $b_n = n^2 - 5n - 6$, the number of terms satisfying $b_n < 0$ is $\mathbf{C}$, and the sum of such terms is $-\mathbf{DE}$.
(3) It follows that for the sequences $\{a_n\}$ and $\{b_n\}$ in (1) and (2),
$$\sum_{k=1}^{n} \frac{k^2 b_k}{a_k} = \frac{1}{\mathbf{F}}\, n(n + \mathbf{G})\left(n^2 - \mathbf{H}\, n - \mathbf{I}\right).$$