Ratio of sums of first n even numbers and n odd numbers will be

English

Gujarati

Hindi

8. Sequences and Series

easy

The ratio of the sums of first $n$ even numbers and $n$ odd numbers will be

A

$1:n$

B

$(n + 1):1$

C

$(n + 1):n$

D

$(n - 1):1$

Solution

(c) Let ${S_{Even}} = 2 + 4 + 6 + 8 + ……..\infty $ …..$(i)$

and ${S_{Odd}} = 1 + 3 + 5 + 7 + 9 + …….\infty $ …..$(ii)$

${S_E} = \frac{n}{2}[4 + (n – 1)2] = \frac{n}{2}(2n + 2) = n(n + 1)$

${S_O} = \frac{n}{2}[2 + (n – 1)2] = \frac{n}{2}(2n)$

Now $\frac{{{S_E}}}{{{S_O}}} = \frac{{(n + 1)}}{n}$

or ${S_E}:{S_O} = (n + 1):n$.

Standard 11

Mathematics

Share

Similar Questions

Let ${a_1},{a_2},{a_3}, \ldots $ be terms of $A.P.$ If $\frac{{{a_1} + {a_2} + \ldots + {a_p}}}{{{a_1} + {a_2} + \ldots + {a_q}}} = \frac{{{p^2}}}{{{q^2}}},p \ne q$ then $\frac{{{a_6}}}{{{a_{21}}}}$ equals

hard

(AIEEE-2006)

Write the first five terms of the sequences whose $n^{t h}$ term is $a_{n}=n(n+2)$

easy

Let $A B C D$ be a quadrilateral such that there exists a point $E$ inside the quadrilateral satisfying $A E=B E=C E=D E$. Suppose $\angle D A B, \angle A B C, \angle B C D$ is an arithmetic progression. Then the median of the set $\{\angle D A B, \angle A B C, \angle B C D\}$ is

normal

(KVPY-2020)

Let $a_n, n \geq 1$, be an arithmetic progression with first term $2$ and common difference $4$ . Let $M_n$ be the average of the first $n$ terms. Then the sum $\sum \limits_{n=1}^{10} M_n$ is

hard

(KVPY-2019)

The sum of the integers from $1$ to $100$ which are not divisible by $3$ or $5$ is

medium