Between 1 and 31,m numbers have been inserted | vedclass

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

Question

(B) Let $A_{1}, A_{2}, \ldots, A_{m}$ be $m$ numbers such that $1, A_{1}, A_{2}, \ldots, A_{m}, 31$ is an $A.P.$Here,$a=1$,$b=31$,and the total number

Vedclass · Accepted Answer

$14$

Vedclass · Answer

(B) Let $A_{1}, A_{2}, \ldots, A_{m}$ be $m$ numbers such that $1, A_{1}, A_{2}, \ldots, A_{m}, 31$ is an $A.P.$Here,$a=1$,$b=31$,and the total number of terms is $n=m+2$.The common difference $d$ is given by $b = a + (n-1)d$,so $31 = 1 + (m+2-1)d$.$30 = (m+1)d \Rightarrow d = \frac{30}{m+1}$ (Equation $1$).The $k^{	ext{th}}$ inserted number is $A_{k} = a + kd = 1 + kd$.Given the ratio of the $7^{	ext{th}}$ and $(m-1)^{	ext{th}}$ inserted numbers is $5:9$:$\frac{1+7d}{1+(m-1)d} = \frac{5}{9}$.Substituting $d = \frac{30}{m+1}$:$\frac{1+7(\frac{30}{m+1})}{1+(m-1)(\frac{30}{m+1})} = \frac{5}{9}$.$\frac{m+1+210}{m+1+30m-30} = \frac{5}{9}$.$\frac{m+211}{31m-29} = \frac{5}{9}$.$9(m+211) = 5(31m-29)$.$9m + 1899 = 155m - 145$.$146m = 2044$.$m = \frac{2044}{146} = 14$.Thus,the value of $m$ is $14$.

Between $1$ and $31$,$m$ numbers have been inserted in such a way that the resulting sequence is an $A.P.$ and the ratio of the $7^{\text{th}}$ and $(m-1)^{\text{th}}$ inserted numbers is $5:9$. Find the value of $m$.

Similar Questions

If the sum of three consecutive terms of an $A.P.$ is $51$ and the product of the last and first term is $273$,then the numbers are

Let $AP(a; d)$ denote the set of all the terms of an infinite arithmetic progression with first term $a$ and common difference $d > 0$. If $AP(1; 3) \cap AP(2; 5) \cap AP(3; 7) = AP(a; d)$,then $a + d$ equals:

If $\sum_{r=1}^n(2r+1)=440$,then $n = \ldots$.

If the sum of $n$ terms of an arithmetic progression is $2n^2 + 5n$,then its $n^{th}$ term is.........

The sum of $n$ terms of two arithmetic progressions are in the ratio $(3n + 8) : (7n + 15)$. Find the ratio of their $12^{\text{th}}$ terms.

Vedclass Test Series

Exam Paper Generator

Online Exam Module