Thenth Term Of A Geometric Sequence

If {a} is a geometric sequence with common ratio_{n}r, then

a_{2}=a_{1}⋅r

a_{3}=a_{2}⋅r=a_{1}⋅r^{2}

a_{4}=a_{3}⋅r=a_{1}⋅r^{3}

a=_{n}a_{1}⋅r^{(n-1)}for everyn>1

Example 1: Finding a Term in a Geometric Sequence

In a geometric sequence, the third term is 24 and the 6th term is 192. Find the 10th term. Solution: Here, *a*_{3}=*a*_{1}⋅*r*^{2}=24 … (1)

and *a*_{6}=*a*_{1}⋅*r*^{5}=192 … (2)

Dividing (2) by (1), we get *r*=2. Substituting *r*=2 in (1), we get a=6.

Hence *a*_{10}=6⋅2^{9}=3072.

Example 2. If the first and tenth terms of a geometric sequence are 1 and 4, find the seventeenth term to three decimal places.

Solution:

First let *n*=10, *a*_{1}=1, *a*_{10}=4 and use the formula *a _{n}*=

*a*

_{1}⋅

*r*

^{(n-1)}to find

*r*.

*a*=

_{n}*a*

_{1}⋅

*r*

^{(n-1)}

*a*

_{10}=1⋅

*r*

^{(10-1)}

4=

*r*

^{9}→

*r*=4

^{(1⁄9)}

Now use the formula *a _{n}*=

*a*

_{1}⋅

*r*

^{(n-1)}again, this time with

*n*=17.

*a*

_{17}=

*a*

_{1}⋅

*r*

^{17}=1(4

^{(1⁄9)})

^{17}=4

^{(17⁄9)}≈13.716

If we know the first term in a geometric progression and the ratio between successive terms, then we can work out the value of any term in the geometric progression . The

nth term is given by

a=_{n}a_{1}⋅r^{(n-1)}

Again, a is the first term andris the ratio. Remember thata_{1}⋅r^{(n-1)}≠(a_{1}⋅r)^{(n-1)}.

Question 1. Given the first two terms in a geometric progression as 2 and 4, what is the 10th term?

Solution:

*a*

_{1}=2;

*r*=4/2=2

Then

*a*

_{10}=2×2

^{9}=1024.

Question 2. Given the first two terms in a geometric progression as 5 and ½, what is the 7th term?

Solution:

Example 3. Find the 20th and *n*th terms of the geometric sequence 5/2, 5/4, ⅝, …

Solution:

The given geometric sequence is 5/2, 5/4, ⅝, … .

Here, *a*_{1}= First term =5/2.

Question 3. (Challenge) The fifth term of a geometric sequence is 1/27 of the eighth term. If the ninth term is 702, what is the eighth term?

Solution:

Given *a*_{5}=*a*_{8}/27 and *a*_{9}=702.

Find the Value of *r*.

Find the Value of

*a*

_{8}.

*a*

_{9}=

*a*

_{8}⋅

*r*

702=

*a*

_{8}⋅3

*a*

_{8}=234

Answer: 234

Example 4: **Finding Terms in a Geometric Sequence**

If the third term of a geometric sequence is -12 and the fourth term is 24, find the first and fifth terms of the sequence.

Solution:

Divide the 4th term by the 3rd term to find the common ratio.

The common ratio is 24/(-12) or -2. Substitute 3 for *n* and -2 for *r* to find the first term.

*a*=

_{n}*a*

_{1}⋅

*r*

^{(n-1)}

*a*

_{3}=

*a*

_{1}⋅(-2)

^{(3-1)}

-12=

*a*

_{1}⋅(-2)

^{2}

-12=4

*a*

_{1}

-3=

*a*

_{1}

The first term is -3. Find the fifth term.

*a*=

_{n}*a*

_{1}⋅

*r*

^{(n-1)}

*a*

_{5}=-3(-2)

^{(5-1)}

=-3(-2)

^{4}

*a*

_{5}=-48

The fifth term is -48.

Example 5: **Determining Terms and the Number of Terms in a Finite Geometric Sequence**

In a finite geometric sequence, *t*_{1}=5 and *t*_{5}=1280

a) Determine *t*_{2} and *t*_{6}.

b) The last term of the sequence is 20, 480. How many terms are in the sequence?

solution:

a) Determine the common ratio.

Use: *t _{n}*=

*t*

_{1}

*r*

^{(n-1)}

Substitute:

*n*=5,

*t*

_{5}=1280,

*t*

_{1}=5

*r*

^{(5-1)}

Simplify.

*r*

^{4}

Divide each side by 5.

*r*

^{4}

Take the fourth root of each side.

*r*

^{2}

*r*=-4 or

*r*=4

There are 2 possible values for

*r*.

(i) When

*r*=-4, then

*t*

_{2}is 5(4)=20

(ii) When

*r*=4, then

*t*

_{2}is 5(-4)=-20

To determine

*t*

_{6}, use:

*t*=

_{n}*t*

_{1}

*r*

^{(n-1)}

(i) Substitute:

*n*=6,

*t*

_{1}=5,

*r*=-4

*t*

_{6}=5(-4)

^{(6-1)}=5(-4)

^{5}=-5120

(ii) Substitute:

*n*=6,

*t*

_{1}=5,

*r*=4

*t*

_{6}=5⋅4

^{(6-1)}=5⋅4

^{5}=5120

So,

*t*

_{2}is -20 or 20, and

*t*

_{6}is -5120 or 5120.

b) Since the last term is positive, use the positive value of *r*.

*t*=

_{n}*t*

_{1}

*r*

^{(n-1)}

Substitute:

*t*=20 480,

_{n}*t*

_{1}=5,

*r*=4

^{(n-1)}

Divide each side by 5.

^{(n-1)}

Use guess and test to determine which power of 4 is equal to 4096.

Guess: 4

^{4}=256

This is too low. Guess: 4

^{6}=4096

This is correct.

So, 4

^{6}=4

^{(n-1)}

Equate exponents.

*n*-1

*n*=7

There are 7 terms in the sequence.

Ratio of two terms of Geometric Sequence

Let the geometric sequence be

a,ar,a_{1}⋅r^{2}, …

Now,

Example 6. The 5th, 8th and 11th terms of a geometric sequence are *p*, *q* and *s*, respectively. Show that *q*^{2}=*p⋅s*.

Solution:

Let a be the first term and *r* be the common ratio of the geometric sequence

According to the given condition,

*a*

_{5}=

*a*

_{1}⋅

*r*

^{(5-1)}=

*a*

_{1}⋅

*r*

^{4}=

*p*… (1)

*a*

_{8}=

*a*

_{1}⋅

*r*

^{(8-1)}=

*a*

_{1}⋅

*r*

^{7}=

*q*… (2)

*a*

_{1}1=

*a*

_{1}⋅

*r*

^{(11-1)}=ar

^{10}=

*s*… (3)

Dividing equation (2) by (1), we obtain

Dividing equation (3) by (2), we obtain

Equating the values of

*r*

^{3}obtained in (4) and (5), we obtain

Thus, the given result has proven.