(a).

At the beggining the budget of the government is  billion per year.

After  year it decreases by , which is same being multiplied by .

 which is  less than .

After  years it again decreases by  which is same being multiplied by .

 which is  less than .

After  years, the amount budgetted is .

(b).

The budget of first  is calculated by using the formula .

Where  is number of years.

If , then .

If , then .

If , then .

If , then .

(c).

Observe the sequence:

The sequence is clearly decreasing and it is always positive meaning that it is bounded below by .

According to bounded monotonic sequences, if the sequence  is bounded and monotonic,then it converges.

Therefore the sequence converges.

Further the limit of the sequence is .

The sequence converges and the limit is .

(a) Formula for the sequence is .

(b)

If , then .

If , then .

If , then .

If , then .

(c) The sequence converges and the limit is .