Question

Two APs have the same common difference. The difference between their 100 th term is 100, what is the difference between their 1000 th terms?

soln:-Let the first term of these A.P.s be a 1 and a 2 respectively and the common difference of these A.P.s be d .

For first A.P.,

a 100=a 1+ (100 − 1)d

=a 1+ 99d

a 1000=a 1+ (1000 − 1)d

a 1000=a 1+ 999d

For second A.P.,

a 100=a 2+ (100 − 1)d

=a 2+ 99d

a 1000=a 2+ (1000 − 1)d

=a 2+ 999d

Given that, difference between

100thterm of these A.P.s = 100

Therefore, (a 1+ 99d) − (a 2+ 99d) = 100

a 1−a 2= 100 (1)

Difference between 1000thterms of these A.P.s

(a 1+ 999d) − (a 2+ 999d) =a 1−a 2

From equation (1),

This difference,a 1−a 2= 100

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

but


Difference between 1000thterms of these A.P.s

(a 1+ 999d) − (a 2+ 999d) =a 1−a 2


how this step happen please explain me ?

Answer 1

Here is another solution....

Answer 2

ANSWER

an=a+(n−1)d100th term will be,a100=a+(100−1)da100=a+99d100th term of 1st Ap = a + 99d2nd AP will be,bn=bn+(n−1)db100=b+99da100−b100=100(a+99d)−(b+99d)=100a−b=100...(1) Difference between 1000th term,=a1000−b1000=a+(1000−1)d)−[b+(1000−1)d]=a−b+999d−999d=100