if sum of 40 A Ms between two numbers is 120,find the sum 50 AMs between them - Maths - Sequences and Series

Question

if sum of 40 A Ms between two numbers is 120,find the sum 50 AMs
between them

Shridhar Kaushik · Accepted Answer

Let A1, A2, A3, , A40 be 40 A M's between two numbers 'a' and 'b' Then, a, A1, A2, A3, , A40, b is an A P with common difference d = $\frac{b - a}{41}$ Now, $A_{1} + A_{2} + + A_{40} = \frac{40}{2} {A_{1} + A_{40}} \Rightarrow A_{1} + A_{2} + + A_{40} = \frac{40}{2} {a + b} [\begin{matrix} ∵ a, A_{1}, A_{2},, A_{40}, b is an A P \\ ∵ a + b = A_{1} + A_{40} \end{matrix}]$ But sum of 40 A M 's is 120 [given] âˆ´ 120 = 20 (a + b) â‡’ (a + b) = 6 (1) Again consider B1, B2, , B50 be 50 A M 's between two numbers a and b Then, a, B1, B2, , B50, b will be in A P with common difference $\frac{b - a}{51}$ Now, $B_{1} + B_{2} + + B_{50} = \frac{50}{2} {B_{1} + B_{50}} = \frac{50}{2} {a + b} = 25 \times 6 = 150$

if sum of 40 A.Ms between two numbers is 120,find the sum 50 AMs between them.