**Question 1.Determine whether the sum to infinity of the following G.P.s exist, if exists find them.**

Solution:

Solution:

Solution:

Solution:

**(v) 9, 8.1, 7.29, ……Solution:**9, 8.1, 7.29, …..

**Question 2.Express the following recurring decimals as rational numbers.**

∴ Sum to infinity exists.

∴ Sum to infinity

∴ Sum to infinity exists.

∴ Sum to infinity

∴ Sum to infinity exists.

∴ Sum to infinity

**Question 3.If the common ratio of a G.P. is 2/3 and the sum to infinity is 12, find the first term.Solution:**

**Question 4.If the first term of the G.P. is 6 and its sum to infinity is 96/17, find the common ratio.Solution:**a = 6, sum to infinity = 96/17 …..[Given]

Sum to infinity = a/1-r

**Question 5.The sum of an infinite G.P. is 5 and the sum of the squares of these terms is 15, find the G.P.Solution:**

Let the required G.P. be a, ar, , , …..

Sum to infinity of this G.P. = 5

**Question 6.FindSolution:**

Solution:

Solution:

Solution:

**Question 7.The midpoints of the sides of a square of side 1 are joined to form a new square. This procedure is repeated indefinitely. Find the sum of(i) the areas of all the squares.(ii) the perimeters of all the squares.Solution:**

and so on.

∴ Sum of the areas of all the squares

∴ Sum to infinity exists.

and so on.

Sum of the perimeters of all the squares

**Question 8.A ball is dropped from a height of 10 m. It bounces to a height of 6m, then 3.6 m, and so on. Find the total distance travelled by the ball.Solution:**

Here, on the first bounce, the ball will go 6 m and it will return 6 m.

On the second bounce, the ball will go 3.6 m and it will return 3.6 m, and so on.

Given that, a ball is dropped from a height of 10 m.

∴ Total distance travelled by the ball is = 10 + 2[6 + 3.6 + …]

The terms 6, 3.6 … are in G.P.

a = 6, r = 0.6, |r| = |0.6| < 1

∴ Sum to infinity exists.

∴ Total distance travelled by the ball