Download Sequences and Series level 1 book 2

Definitions:

A progression of numbers is called a sequence. E.g., 2, 4, 6, 8,…, 100. The
sequence can be finite if it ends or infinite if it is unending. In our example,
the sequence ends with the 50th term, 100, and is therefore a finite sequence.
Sequences and Series Level 1 Book 1
2

A term of a sequence (or series) is an individual number in the progression.
In our example above, the 3rd term is 6.

A Series is the sum of a progression of numbers. E.g., 2+4+6+8+…+100.
Often we are interested in the particular number that this sum equals, in our
example we can most easily find the sum if we rearrange the addition:
(2+100)+(4+98)+ … + (50+52). In this way we see that the sum is half the
number of terms, 25, times the sum of the first and last term, 102= 2550.

Differences in a sequence are the numbers obtained by subtracting a term
from its successive term. In our example, the differences are:
 4-2 = 2
 6-4 = 2
 etc.

An Arithmetic Sequence is a sequence in which all the differences are the
same. Our example 2, 4, 6,…, 100 is an arithmetic sequence because the
differences all equal 2. A second example, 1, 3, 6, 10,…, 45 is not an arithmetic
sequence because its differences for the sequence 1, 2, 3,…, 9 and hence the
differences aren’t all the same.

An Arithmetic Series is the sum of an arithmetic sequence.

In an arithmetic sequence (or series,) the constant difference of consecutive
terms is called the Delta.

The General Term of a sequence (or series) is a kind of basket that represents
all of the terms of the sequence at once. In our example 2, 4, 6, … the general
term is {2x}. When x=3, we see that 2x=6 is the third term. The general term
makes it easy to predict that the 100th term is 200 or that the 1034th term is
2068.
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Sequences and Series Level 1 Book 1
Technique for Arithmetic Series
1. Find the delta or common difference of all the terms. This identifies the times
table that your series is a variation on, and hence tells you what to multiply x
by in the general term. In the example 3+5+7+9+ …+97 this delta would be 2
telling us that this sequence is a variation on the 2-times table series:
2+4+6+8+…+96
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+5+7+9+…+97. This would be 1.
This is what must be added to the multiple of x (2) found in step 1.
3. Put the above two parts together and you get the general term: {2x+1}.
4. Now you are in a position to use the general term to make predictions. For
instance, the 1000th term is 2001. We can also find out what term number is
97 by solving the equation 2x+1=97 to find that this is term number 48.
5. Now rearrange the terms: (3+97)+(5+95)+(7+93)+…+(49+51) where you can
easily see that our sum equals 100 * 24 (24=half the number of terms—
needed since we pair them up.) Sum=2400.
Find the sum of the series:
11+15+19+23+…+207
Delta:
4
Term 0:
7
General Term:
4x + 7
What term number is 207:
50
Find the sum of the end terms:
Find the sum of the series:
218 = 5450
Find the sum of the series:
4+11+18+25+…+137
Delta:
7
Term 0:
-3
General Term: 7x - 3
What term number is 137:
20
218
25 *
Sequences and Series Level 1 Book 1
Find the sum of the end terms:
Find the sum of the series:
141 = 1410
4
141
10 *
5
Sequences and Series Level 1 Book 1
Find the sum of 10 terms of the series:
3+9+15+21+…
Delta:
Term 0:
General Term:
Term 10:
Find the sum of the end terms:
Find the sum of the series:
Find the sum of the series:
20+30+40+50+…+120
Delta:
Term 0:
General Term:
What term number is 120:
Find the sum of the end terms:
Find the sum of the series:
Find the sum of 70 terms of the series:
12+16+20+24+…
Delta:
Term 0:
General Term:
Term 70:
Sequences and Series Level 1 Book 1
Find the sum of the end terms:
Find the sum of the series:
Find sum of 10 terms of the series:
7+8+9+10+…
Delta:
Term 0:
General Term:
Term 10:
Find the sum of the end terms:
Find the sum of the series:
Find sum of 60 terms of the sequence
9+12+15+18+…
6
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Sequences and Series Level 1 Book 1
Find sum of the series
12+19+26+33+…+705
Sum the series 1+9+17+25+…+393
Sum 16 terms of the series:
19+27+35+43+…
Sequences and Series Level 1 Book 1
Sum the series: 1+2+3+…+1000
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Sequences and Series Level 1 Book 1
New Skill Introduced—Summation Notation
Mathematicians use the capital Greek letter  to indicate summation.
Hence,
100
 2n  2  4  6  ...  200
n 1
In the following examples, expand the summation to find the series you are
working with. Then sum the series as you have done before.
Find
100
 (2n  1)
n 1
Actual Series to be summed: 1+3+5+…+199
50
Find
 (3n  2)
n 1
Actual Series to be summed:
Sequences and Series Level 1 Book 1
63
 (5n  8)
n 1
Find
40
 (8n  7)
n 1
Find
30
 (3n  1)
n 1
Find
10
11
Sequences and Series Level 1 Book 1
11
 ( 4 n  6)
n 1
Find
100
 (10n  4)
n 1
Find
20
 (4n  5)
n 1
Sequences and Series Level 1 Book 1
Find
63
 (5n  2)
n 1
Find
30
 ( 4 n  9)
n 1
Find
12
13
Sequences and Series Level 1 Book 1
Find
Answers
Page
No.
3
4
5
Exampl
e No.
1
2
1
2
3
1
2
3
Delta
4
7
6
10
4
1
3
7
Term
0
7
-3
-3
10
8
6
6
5
General Term or
Expanded Series
4x+7
7x-3
6x-3
10x+10
4x+8
x+6
3x+6
7x+5
nth Term
n=50
n=20
57
n=11
288
16
186
n=100
Sum of
End Terms
218
141
60
140
300
23
195
717
Series
Sum
5450
1410
300
770
10500
115
5850
35850
-7
11
0
-1
2
8
-7
1
6
4
8x-7
8x+11
x
1+3+5+…
5+8+11+…
13+18+23+…
1+9+17+…
4+7+10+…
10+14+18+…
114+24+34+…
n=50
139
n=1000
199
152
323
313
91
50
1004
384
158
1001
200
157
336
314
95
60
1018
9850
1264
500500
10000
3925
10584
6280
1425
330
50900
60
 ( 6 n  9)
n 1
6
7
8
9
1
2
3
1
2
1
2
3
1
2
8
8
1
2
3
5
8
3
4
10
Sequences and Series Level 1 Book 1
10
3
1
2
3
4
4
4
6
5
6
9
9
9+13+17+…
7+12+17+…
13+17+21+…
15+21+27+…
85
317
129
369
14
94
324
142
384
940
10206
2130
11520
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