Sequences and Series Level 1 Book 1
2
Definitions:






A progression of numbers is called a sequence. E.g., 2, 4, 8, 16,…, 1024 The
sequence can be finite if it ends or infinite if it is unending. In our example,
the sequence ends with the 10th term, 1024, and is therefore a finite sequence.
A term of a sequence (or series) is an individual number in the progression.
In our example above, the 3rd term is 8.
A Series is the sum of a progression of numbers. E.g., 2+4+8+…+1024. Often
we are interested in the particular number that this sum equals.
Ratios in a sequence are the numbers obtained by dividing a term by its
previous term. In our example, the ratios are:
 4/2=2
 8/4=2
 etc.
An Geometric Sequence is a sequence in which all the ratios are the same.
Our example 2, 4, 8, …, 1024 is a geometic sequence because the ratios all
equal 2. A second example, 1, 3, 5, 7, …, 45 is not a geometic sequence
because its ratios aren’t all the same.
The General Term of a sequence is a kind of basket that represents all of the
terms of the sequence at once. In our example 2, 4, 8, … the general term is
{2x }. When x=3, we see that 23 = 8 is the third term. The general term makes
it easy to predict that the 10th term is 210 = 1024 or that the 1034th term is
21034.
Technique for Geometic Sequences
1. Find the ratio. This identifies the power (exponential) function that your
sequence is a variation on, and hence tells you what base to raise to the x
power. In the example 3, 6, 12, 24, … this ratio would be 2 telling us that this
sequence is a variation on the 2x function: 2, 4, 8, 16,…
2. Find what term zero of the sequence would be. In our example above, term
zero is what would come before the 3 in 3, 6, 12, 24,…. This would be 1.5.
This is what must be multiplies by the power function of x found in step 1.
3. Put the above two parts together and you get the general term: {1.5(2x)}.
4. Now you are in a position to use the general term to make predictions. For
instance, the 10th term is 1.5(210) = 1.5(1024) = 1536. We can also find out
what term number is 192 by solving the equation 1.5(2x)=192 to find that
2x=128 and hence x=7.
3
Sequences and Series Level 1 Book 1
Technique for Summing a Geometric Series
 Consider the simple geometric series 3+6+12+…+3072.
 Find the ratio: r=6/3 = 2.
 Represent the sum by the letter S: S=3+6+…+3072.
 Notice what rS equals: rS=6+12+…+3072+6144.
 All the middle terms of S and of rS are the same—they differ only in their end
terms. Hence, rS-S contains only 2 terms. These are 6144, which is r times
the last term, and 3 which is the first term. But when you subtract one S from
r S’s, you get r-1 of them. This gives the essential idea presented next.

2  3072  3
 6141  In summary, the sum is found by dividing (r
2 1
times
last term – first term) by (r-1).
S
Find sum of the series
10+20+40+80+…+5120
Ratio:
2
Ratio times the last term:
Ratio - 1:
1
Sum of the Series:
10230
10240
(10240-10)/1 =
Find sum of the series
6+18+54+162+…+1458
Ratio:
3
Ratio times the last term:
Ratio - 1:
2
4374
Sequences and Series Level 1 Book 1
Sum of the Series:
2184
(4374-6)/2 =
4
5
Sequences and Series Level 1 Book 1
Find the sum of the series:
20+200+2000+…+2000000
Ratio:
Ratio times the last term:
Ratio - 1:
Sum of the Series:
Find sum of the series
3+9+27+81+…+2187
Ratio:
Ratio times the last term:
Ratio - 1:
Sum of the Series:
Find the 11th term of the sequence
48+24+12+…+3
Ratio:
Ratio times the last term:
Sequences and Series Level 1 Book 1
Ratio - 1:
Sum of the Series:
Find sum of the series
1728+576+192+…+64
Ratio:
Ratio times the last term:
Ratio - 1:
Sum of the Series:
Find the 10th term of the sequence
1+5+25+125+…+1953125
Ratio:
Ratio times the last term:
Ratio - 1:
Sum of the Series:
6
7
Sequences and Series Level 1 Book 1
Find sum of the series
81+54+36+…+(64/243)
Find sum of the series
+ … + 4096/3
1/3 + 2/3 + 4/3
Find the sum of the series
1/4, 1/8 +…+ 1/1024
1 + 1/2 +
Sequences and Series Level 1 Book 1
Find the sum of the series
2 + 8 +…+32768
8
1/8 + 1/2 +
Answers
Page
No.
Example
No.
Ratio
Ratio times last
term
Ratio
minus one
Sum of the Series
3
1
2
10240
1
10230
2
3
4374
2
2184
1
10
20000000
9
2222220
2
3
6561
2
3279
4
9
Sequences and Series Level 1 Book 1
5
6
3
1/2
3/2
-1/2
93
1
1/3
64/3
-2/3
2560
2
5
9765625
4
2441406
3
2/3
128/729
-1/3
58921/243
1
2
8192/3
1
8191/3
2
1/2
1/2048
-1/2
2047/1024
3
4
131072
3
349525/8
Green Valley School
389 Pembroke Street
Pembroke, NH 03275
(603) 485-8550