**How to Find the Geometric Progression with Given Information ?**

Here we are going to see some practice questions of finding geometric progression with given information.

**Question 1 :**

In a G.P. the 9^{th} term is 32805 and 6^{th} term is 1215. Find the 12^{th} term.

**Solution :**

9 t ar |
6 t ar |

(2) / (1) ==> ar^{5}/ar^{8} = 1215/32805

1/r^{3} = 1/27

(1/r)^{3} = (1/3)^{3}

r = 3

By applying the value of r in (2), we get

a(3)^{5} = 1215

a = 1215/3^{5}

a = 5(3^{5})/3^{5}

a = 5

12th term :

t_{12 } = ar^{11}

t_{12 } = 5(3)^{11}

**Question 2 :**

Find the 10^{th} term of a G.P. whose 8^{th} term is 768 and the common ratio is 2.

**Solution :**

8^{th} term is 768

t_{8} = 768

ar^{7} = 768

r = 2

a(2^{7}) = 768

a = 768/128

a = 6

10th term :

t_{10} = ar^{9}

= 6(2^{9})

= 6(512)

t_{10 }= 3072

Hence the 10^{th} term of the sequence is 3072.

**Question 3 :**

If a, b, c are in A.P. then show that 3^{a}, 3^{b}, 3^{c} are in G.P.

**Solution :**

**If a, b, c are in A.P, then b = (a + c)/2**

**In G.P, b = **√ac

**To prove **that 3^{a}, 3^{b}, 3^{c} are in G.P.

3^{b}** = **√(3^{a }⋅ 3^{c})

3^{b}** = (**3^{a + c})^{1/2}

3^{b}** = **3^{(a+c)/2}

b = (a + c)/2

Hence 3^{a}, 3^{b}, 3^{c }are in G.P.

**Question 4 :**

In a G.P. the product of three consecutive terms is 27 and the sum of the product of two terms taken at a time is 57/2 . Find the three terms.

**Solution :**

Let the three terms be a/r, a, ar

The product of three consecutive terms = 27

(a/r) ⋅ a ⋅ ar = 27

a^{3} = 27

a = 3

Sum of the product of two terms taken at a time = 57/2

[(a/r) ⋅ a] + [a ⋅ ar] + [ar ⋅ a/r] = 57/2

a^{2}/r + a^{2}r + a^{2} = 57/2

a^{2}(1/r + r + 1) = 57/2

9(1 + r + r^{2})/r = 57/2

18(r^{2 }+ r + 1) = 57 r

18r^{2 }+ 18r + 18 = 57 r

18r^{2 }+ 18r - 57r + 18 = 0

18r^{2 } - 39r + 18 = 0

6r^{2 } - 13r + 6 = 0

(2r - 3)(3r - 2) = 0

r = 3/2 and r = 2/3

First term = a/r = 3/(3/2) = 2

Second term = a = 3 = 3

Third term = ar = 3(3/2) = 9/2

Hence the required three terms are 2, 3, 9/2 (or) 9/2, 3, 2.

After having gone through the stuff given above, we hope that the students would have understood, "How to Find the Geometric Progression with Given Information".

Apart from the stuff given in this section "How to Find the Geometric Progression with Given Information", if you need any other stuff in math, please use our google custom search here.

Widget is loading comments...