Sequences and Series · Mathematics · AP EAPCET

If $S_n=1^3+2^3+\ldots+n^3$ and $T_n=1+2+\ldots+n$, then

$\frac{1}{3 \cdot 5}+\frac{1}{5 \cdot 7}+\frac{1}{7 \cdot 9}+\ldots$ to 24 terms $=$

$$ 1+\frac{4}{15}+\frac{4 \cdot 10}{15 \cdot 30}+\frac{4 \cdot 10 \cdot 16}{15 \cdot 30 \cdot 45}+\ldots . .+\infty= $$

If $t_n=\frac{1}{4}(n+2)(n+3), n \in N$, then which one of the following is true?

Assertion (A) $\frac{1}{t_1}+\frac{1}{t_2}+\ldots+\frac{1}{t_{2003}}=\frac{2003}{3009}$

Reason (R) $\frac{1}{t_1}+\frac{1}{t_2}+\ldots+\frac{1}{t_n}=\frac{4 n}{(2 n+3)}$

The sum of all integers between 1 and 100 (both inclusive) which are divisible by 5 or 13 is

If $x>\sqrt{3}$ and $\frac{x^2+1}{\left(x^2+2\right)\left(x^2+3\right)}$ is expanded in terms of powers of $x$, then the coefficient of $x^{-8}$ is

$$ \sum\limits_{k=1}^n k(k+1)(k+2) \ldots(k+r-1)= $$

For all $n \in N, \frac{3^n-1}{2} \geq$

If $2 \cdot 5+5 \cdot 9+8 \cdot 13+11 \cdot 17+\ldots$ to $n$ terms $=a n^3+b n^2+c n+d$, then $a-b+c-d=$

$$ 2+3+5+6+8+9+\ldots .2 n \text { terms }= $$

Suppose that the three points $$A, B$$ and $$C$$ in the plane are such that their $$x$$-coordinates as well as $$y$$-coordinates are in GP with the same common ratio. Then, the points $$A, B$$ and $$C$$

Using mathematical induction, the numbers $$a_n^{\prime}$$ s are defined by $$a_0=1, a_{n+1}=3 n^2+n+a_n (n \geq 0)$$, then $$a_n$$ is equal to

If $$1+x^2=\sqrt{3} x$$, then $$\sum_{n=1}^{24}\left(x^n-\frac{1}{x^n}\right)^2$$ is equal to

Let $$p$$ and $$q$$ be the roots of the equation $$x^2-2 x+A=0$$ and let $$r$$ and $$s$$ be the roots of the equation $$x^2-18 x+B=0$$. If $$p < q < r < s$$ are in AP then the values of $$A$$ and $$B$$ are

Let $$f(x)=x^3+a x^2+b x+c$$ be polynomial with integer coefficients. If the roots of $$f(x)$$ are integer and are in Arithmetic Progression, then $$a$$ cannot take the value