Question

Verify if the sequence of numbers formed will form an AP or not.


1- The amount of air present in the cylinder when a vacuum pump removes each time 1/4 of air remaining in the cylinder.
2- Divya deposited ₹1000 at compound interest at the rate of 10% per annum. The amount at the end of first year, second year, third year,....., and so on.

Answer 1

1)
Let us assume, initial quantity of air = V

Therefore, quantity removed in the first step = V × 14

Quantity remaining after the first step

= V−V4 = 3V4

Quantity removed in the second step = 3V4 × 14

= 3V16

Quantity remaining after the second step

= 3V4 - 3V16 = 9V16

Quantity remaining after each step follows the trend:

V,3V4,9V16,...

Here each subsequent term is not obtained by adding a fixed number to the previous term.

Hence, it is not an AP.

2)
First year principal = A0 = Rs 1000

r = 10%

Time period ( t ) = 1 yr
Amount at the end of

First year( A1) = 1000 ( 1 + 10 / 100 )


= 1000 × 110 / 100

A1 = Rs 1100

second year principal = A1 = Rs 1100

r = 10 %

Time period ( t ) = 1 yr
Amount at the end of

Second year( A2) = 1100 ( 1 + 10 / 100 )

= 1100 × 110 /100

A2 = Rs 1210

third year principal = A2 = Rs 1210

r = 10 %

Time period ( t ) = 1 year

A3 = 1210 ( 1 + 10 /100 )

= 1210 × 110 / 100

A3 =Rs 1331

Therefore,

First year , second year ,third year ending Amounts are

1100 , 1210 , 1331 ,...

a2 - a1 = 1210 - 1100 = 110

a3 - a2 = 1331 - 1210 = 121

Therefore ,

a2 - a1 is not equal to ( a3 - a2 )

This is not an A.P

I hope this helps you.