SOLUTION: Use the geometric sequence of numbers 1, 2, 4, 8,... to find the following: a) What is r, the ratio between 2 consecutive terms? b) Using the formula for the nth term of a ge

Use the geometric sequence of numbers 1, 2, 4, 8,...
to find the following: 

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a) What is r, the ratio between 2 consecutive terms? 


Just divide each given term  after the first by the 
preceding one and see if you get the same number.  If 
you do, then you call that number "the common ratio, r".

For 1, 2, 4, 8,... we divide the second term, 2, by the 
first term 1, like this: 2÷1 = 2.  Then we divide the 
third term 4, by the second term 2, like this: 4÷2 = 2. 
Then we divide the fourth term, 8, by the third term, 4, 
like this" 8÷4 = 2.  

Every time we got 2. So that means this is a geometric 
sequence and the common ratio, r, is 2.  So r = 2. 

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b) Using the formula for the nth term of a geometric 
sequence, what is the 24th term? 

The formula for the nth term, called an, of a geometric 
sequence is 

an = a1rn-1

where a1 stands for the first term, r stands for the 
common ratio, and n stands for the number of term that 
you want to find.
 
Here a1 = 1, r = 2, and n = 24 so we plug those in:

an = a1rn-1

a24 = (1)(2)(24)-1

a24 = 223 = 8388608

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c) Using the formula for the sum of a geometric series, 
what is the sum of the first 10 terms?

The formula for the sum, called Sn, of the first n terms 
of a geometric sequence is either of these two equivalent 
formulas:

Sn = a1(rn - 1)/(r - 1)

or

Sn = a1(1 - rn)/(1 - r)

where a1 stands for the first term, r stands 
for the common ratio, and n stands for the number of 
term that you want to find.  It doesn't matter which of 
those formulas you use, because you'll get the same 
answer using either one. Normally we use the first one 
if |r| > 1 and the second one if |r| < 1, but there is 
no rule. I'll use the first one. 
 
Here a1 = 1, r = 2, and n = 10 so we plug those in:

Sn = a1(rn - 1)/(r - 1)

S10 = (1)(210 - 1)/(2 - 1)

S10 = (210 - 1)/1

S10 = 210 - 1

S10 = 1024 - 1

S10 = 1023

Edwin