Question

two APs have the same common difference.the difference between their 100th tems is 100,what is the difference between their 1000th term?

Answer 1

Let the first term of these A.P.s be a1 and a2 respectively and the common difference of these A.P.s bed.
For first A.P.,
a100 = a1 + (100 − 1) d
= a1 + 99d
a1000 = a1 + (1000 − 1) d
a1000 = a1 + 999d
For second A.P.,
a100 = a2 + (100 − 1) d
= a2 + 99d
a1000 = a2 + (1000 − 1) d
= a2 + 999d
Given that, difference between
100th term of these A.P.s = 100
Therefore, (a1 + 99d) − (a2 + 99d) = 100
a1 − a2 = 100 ... (i)
Difference between 1000th terms of these A.P.s
(a1 + 999d) − (a2 + 999d) = a1 − a2
From equation (i),
This difference, a1 − a2 = 100
Hence, the difference between 1000th terms of these A.P. will be 100.

Answer 2

Let the first term of these A.P.s be a1 and a2 respectively and the common difference of these A.P.s be d.

For first A.P.,

a100 = a1 + (100 − 1) d

= a1 + 99d

a1000 = a1 + (1000 − 1) d

a1000 = a1 + 999d

For second A.P.,

a100 = a2 + (100 − 1) d

= a2 + 99d

a1000 = a2 + (1000 − 1) d

= a2 + 999d

Given that, difference between

100th term of these A.P.s = 100

Therefore, (a1 + 99d) − (a2 + 99d) = 100

a1 − a2 = 100 (1)

Difference between 1000th terms of these A.P.s

(a1 + 999d) − (a2 + 999d) = a1 − a2

From equation (1),

This difference, a1 − a2 = 100

Hence, the difference between 1000th terms of these A.P. will be 100.



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