Let l_1, l_2, ldots, l_{100} be consecutive t | vedclass

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

Question

(A) Given $A_{51} - A_{50} = 1000$. Since $l_i = l_1 + (i-1)d_1$ and $w_i = w_1 + (i-1)d_2$,we have $A_i = l_i w_i = (l_1 + (i-1)d_1)(w_1 + (i-1)d_2)$

Vedclass · Accepted Answer

$18900$

Vedclass · Answer

(A) Given $A_{51} - A_{50} = 1000$. Since $l_i = l_1 + (i-1)d_1$ and $w_i = w_1 + (i-1)d_2$,we have $A_i = l_i w_i = (l_1 + (i-1)d_1)(w_1 + (i-1)d_2)$. $A_{51} - A_{50} = (l_1 + 50d_1)(w_1 + 50d_2) - (l_1 + 49d_1)(w_1 + 49d_2) = 1000$. Expanding this,we get $l_1 d_2 + w_1 d_1 + (50^2 - 49^2)d_1 d_2 = 1000$. Since $d_1 d_2 = 10$,this becomes $l_1 d_2 + w_1 d_1 + 99(10) = 1000$,which simplifies to $l_1 d_2 + w_1 d_1 = 10$. Now,$A_{100} - A_{90} = (l_1 + 99d_1)(w_1 + 99d_2) - (l_1 + 89d_1)(w_1 + 89d_2)$. $= (l_1 d_2 + w_1 d_1)(99 - 89) + (99^2 - 89^2)d_1 d_2$. $= 10(10) + (99 - 89)(99 + 89)(10) = 100 + 10(188)(10) = 100 + 18800 = 18900$.

Similar Questions

If the roots of the equation $4x^3 - 12x^2 + 11x + k = 0$ are in arithmetic progression,then $k$ is equal to

If $a, b, c, d, e, f$ are in $A.P.$,then the value of $e - c$ is

If the sum of $p$ terms of an arithmetic progression is equal to the sum of its $q$ terms,then what is the sum of its $(p + q)$ terms?

Consider the real-valued function $h: \{0, 1, 2, \ldots, 100\} \rightarrow \mathbb{R}$ such that $h(0) = 5$, $h(100) = 20$, and satisfying $h(p) = \frac{1}{2}\{h(p+1) + h(p-1)\}$ for every $p = 1, 2, \ldots, 99$. Then the value of $h(1)$ is:

The sum of three terms of an Arithmetic Progression $(AP)$ is $18$ and the sum of their squares is $158$. The largest term is.......

Vedclass Test Series

Exam Paper Generator

Online Exam Module