Geometric Series

Another known series is the geometric series also called as Geometric progression. The following term of the series is obtained by multiplying a constant multiplier called a common ratio, equivalent to the common difference of arithmetic progression. It is also an advantage that we are equipped with our knowledge in other topic especially evaluating exponential equations. That skill will help us solve harder problem about this type of series.

Formula derivation:

Nth term                                             Formula

1st term                                                 a_1

2nd term                                               a_1\cdot r

3rd term                                                a_1\cdot r^2

4th term                                                a_1\cdot r^3

5th term                                                 a_1\cdot r^4

Now, observe how the nth term of the series and the power of r related with each other.

Without going further to 6th term, 7th term to nth term we can solve it using the following formula.

a_n=a_1\cdot r^{n-1}

where a_n– is the nth term, a_1– is the first term, n– is the number of terms.


Worked Problem 1:

Find the 8th term of the series {4,8,16…}.


Using the our formula,

a_n=a_1\cdot r^{n-1} a_8=4\cdot 2^{8-1} a_8=4\cdot 2^7 a_8=512


Worked Problem 2:

The 2nd term of a geometric progression is 8 and the 7th term is \displaystyle\frac{1}{4}. What is the first term?


a_n=a_1\cdot r^{n-1} can be written as a_n=a_m\cdot r^{n-m}

a_7=a_2\cdot r^{7-2} \displaystyle\frac{1}{4}=8\cdot r^5 \displaystyle\frac{1}{4\cdot 8}=r^5 \displaystyle\frac{1}{32}=r^5 r=\displaystyle\frac{1}{2}

Since the second term is 8 the previous number is the first term and can be obtained by multiplying the second term by 2. Making 16 is the first term.


Worked Problem 3:

The nth term of geometric progression is given by the equation a_n=3^{n-1}. Find the 5th term.


a_n=3^{n-1} a_5=3^{5-1} a_5=81

Worked Problem 4:

The terms 4,x,25 forms a geometric sequence. What is the value of x?


Let r be the common ration.

4r_1=x, also xr_2=25

r_1=r_2 \displaystyle\frac{x}{4}=\displaystyle\frac{25}{x} x^2=100 x=10

By the way, x is also called the geometric mean.

GM=\sqrt{ab} GM=\sqrt{4\cdot 25} GM=\sqrt{100} GM=10











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