**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".

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