If A is an arithmetic mean between two number | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let the two numbers be $a$ and $b$.The arithmetic mean $A$ between $a$ and $b$ is given by $A = \frac{a+b}{2}$.Let $A_1, A_2, \dots, A_n$ be the $

Vedclass · Accepted Answer

$S = nA$

Vedclass · Answer

(A) Let the two numbers be $a$ and $b$.The arithmetic mean $A$ between $a$ and $b$ is given by $A = \frac{a+b}{2}$.Let $A_1, A_2, \dots, A_n$ be the $n$ arithmetic means between $a$ and $b$. Then $a, A_1, A_2, \dots, A_n, b$ form an arithmetic progression $(A.P.)$ with $n+2$ terms.The common difference $d$ is given by $d = \frac{b-a}{n+1}$.The sum $S$ of $n$ arithmetic means is $S = A_1 + A_2 + \dots + A_n$.Since $A_1, A_2, \dots, A_n$ are in $A.P.$,their sum is $S = \frac{n}{2}(A_1 + A_n)$.Here,$A_1 = a + d = a + \frac{b-a}{n+1} = \frac{an + a + b - a}{n+1} = \frac{an + b}{n+1}$ and $A_n = b - d = b - \frac{b-a}{n+1} = \frac{bn + b - b + a}{n+1} = \frac{bn + a}{n+1}$.Thus,$S = \frac{n}{2} \left( \frac{an + b + bn + a}{n+1} \right) = \frac{n}{2} \left( \frac{(a+b)(n+1)}{n+1} \right) = \frac{n(a+b)}{2}$.Since $A = \frac{a+b}{2}$,we have $S = nA$.

If $A$ is an arithmetic mean between two numbers and $S$ is the sum of $n$ arithmetic means between the same two numbers,then:

Similar Questions

$A$ $G.P.$ consists of an even number of terms. If the sum of all the terms is $5$ times the sum of the terms occupying odd places,then the common ratio will be equal to

Write the seventh term of the following $GP$: $39366, 13122, 4374, 1458, \ldots$

Statement $-1$: The sum of the series $1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16) + \dots + (361 + 380 + 400)$ is $8000$.
Statement $-2$: $\sum_{k=1}^{n} (k^3 - (k-1)^3) = n^3$,for any natural number $n$.

Determine the $25^{th}$ term of an $A.P.$ whose $9^{th}$ term is $-6$ and the common difference is $\frac{5}{4}$.

If $3 + \frac{1}{4} (3 + d) + \frac{1}{4^2} (3 + 2d) + \dots \infty = 8$,then the value of $d$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module