Question

Two terms of AP's have same common difference between their 100th term is 100.Then what is difference between their 1000th term.

Answer 1

a100 = a₁ + (100 − 1) d
⇒a₁ + 99d
a1000 = a₁ + (1000 − 1) d
a1000 = a₁ + 999d
for second ap
a100 = a₂ + (100 − 1) d
⇒ a₂ + 99d
a1000 = a₂ + (1000 − 1) d
⇒ a₂ + 999d
 100th term of these a.p.s = 100
∴ (a₁ + 99d) − (a₂ + 99d) = 100
a₁ − a₂ = 100 ... 1 
∴(a₁ + 999d) − (a₂ + 999d) = a₁ − a₂
from equation 1 
⇔a₁-a₂=100
∴ 100

Answer 2

the common difference would be constant through the AP.

A100 - a100 = 100 

A + 99d - ( a + 99d ) = 100
  
A- a = 100 

similalry

A1000  -a 1000 = x

A +99d - (a +99d) = x 
A - a = X 

x = 100

# hope it helps!!