Further Pure 1 Summation of finite Series. Sigma notation In the last lesson we met the following...
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Transcript of Further Pure 1 Summation of finite Series. Sigma notation In the last lesson we met the following...
![Page 1: Further Pure 1 Summation of finite Series. Sigma notation In the last lesson we met the following rules. 1)1 + 2 + 3 + …… + n = (n/2)(n+1) 2)1 2 + 2 2.](https://reader036.fdocuments.net/reader036/viewer/2022071806/56649db35503460f94aa2803/html5/thumbnails/1.jpg)
Further Pure 1
Summation of finite Series
![Page 2: Further Pure 1 Summation of finite Series. Sigma notation In the last lesson we met the following rules. 1)1 + 2 + 3 + …… + n = (n/2)(n+1) 2)1 2 + 2 2.](https://reader036.fdocuments.net/reader036/viewer/2022071806/56649db35503460f94aa2803/html5/thumbnails/2.jpg)
Sigma notation In the last lesson we met the following rules.
1) 1 + 2 + 3 + …… + n = (n/2)(n+1)
2) 12 + 22 + 32 + …… + n2 = (n/6)(n+1)(2n+1)
3) 13 + 23 + 33 + …… + n3 = (n2/4) (n+1)2
We can write long summations like the ones above using sigma notation.
n
1r
33333
n
1r
22222
n
1r
rn.......321
rn.......321
rn.......321
![Page 3: Further Pure 1 Summation of finite Series. Sigma notation In the last lesson we met the following rules. 1)1 + 2 + 3 + …… + n = (n/2)(n+1) 2)1 2 + 2 2.](https://reader036.fdocuments.net/reader036/viewer/2022071806/56649db35503460f94aa2803/html5/thumbnails/3.jpg)
Sigma notation The r acts as a counter starting at 1 (or whatever is stated under
the sigma sign) and running till you get to n (on top of the sigma sign).
Each r value generates a term and then you simply add up all the terms.
The terms in the example above come fromr = 1 2×1+1 = 3r = 2 2×2+1 = 5r = 3 2×3+1 = 7r = 4 2×4+1 = 9
The 4 on top of the sigma sign tells us to stop when r = 4.
2497531r24
1r
![Page 4: Further Pure 1 Summation of finite Series. Sigma notation In the last lesson we met the following rules. 1)1 + 2 + 3 + …… + n = (n/2)(n+1) 2)1 2 + 2 2.](https://reader036.fdocuments.net/reader036/viewer/2022071806/56649db35503460f94aa2803/html5/thumbnails/4.jpg)
Questions Here are some questions for you to try and find the
values of.
238130683010 )rr(
75302520 r5
7529201385 )4r(
5
2r
3
6
4r
5
1r
2
![Page 5: Further Pure 1 Summation of finite Series. Sigma notation In the last lesson we met the following rules. 1)1 + 2 + 3 + …… + n = (n/2)(n+1) 2)1 2 + 2 2.](https://reader036.fdocuments.net/reader036/viewer/2022071806/56649db35503460f94aa2803/html5/thumbnails/5.jpg)
Sigma notation We can now remember the identities that we met
last lesson and have mentioned already adding the sigma notation.
22n
1r
33333
n
1r
22222
n
1r
)1n(4
nrn.......321
)1n2)(1n(6
nrn.......321
)1n(2
nrn.......321
![Page 6: Further Pure 1 Summation of finite Series. Sigma notation In the last lesson we met the following rules. 1)1 + 2 + 3 + …… + n = (n/2)(n+1) 2)1 2 + 2 2.](https://reader036.fdocuments.net/reader036/viewer/2022071806/56649db35503460f94aa2803/html5/thumbnails/6.jpg)
Using Nth terms
Use the nth term to find the following summation.
The summation only works if you sum from 1 to n. How would you calculate the next example.
Here the sum goes from r = 4, to r = 8. This means you do not want the terms for r = 1, 2 & 3. So the answer will be the sum to 8 minus the sum to 3.
190)13(4
3)18(
4
8rrr
91)162)(16(6
6)1n2)(1n(
6
nr
22
223
1
38
1
38
4
3
6
1
2
![Page 7: Further Pure 1 Summation of finite Series. Sigma notation In the last lesson we met the following rules. 1)1 + 2 + 3 + …… + n = (n/2)(n+1) 2)1 2 + 2 2.](https://reader036.fdocuments.net/reader036/viewer/2022071806/56649db35503460f94aa2803/html5/thumbnails/7.jpg)
Rules of summing series
)1n(2
n5r5
n)...............325(1
n5...............15105r5
n
1r
n
1r
n717
1)...............117(1
7...............7777
n
1r
n
1r
)1n(2
anra ar
n
1r
n
1r
knrk kn
1r
n
1r
Here are 2 rules that you need to be familiar with. There is a numerical example followed by a general rule k and a represent random constants.
![Page 8: Further Pure 1 Summation of finite Series. Sigma notation In the last lesson we met the following rules. 1)1 + 2 + 3 + …… + n = (n/2)(n+1) 2)1 2 + 2 2.](https://reader036.fdocuments.net/reader036/viewer/2022071806/56649db35503460f94aa2803/html5/thumbnails/8.jpg)
Example
3))(n21)(n(n4
n
)6n1)(n(n4
n
6]1)1)[n(n(n4
n
1)(n2
3n1)(n
4
n
r3r3r)(r
2
22
n
1r
n
1r
3n
1r
3
These results can be used to find the sum to n of lots of different series.
First break the summation up.
Next use the general formula.
Here (n/4)(n+1) is a factor Next just multiply out and
collect up like terms. Finally the expression will
factorise.
![Page 9: Further Pure 1 Summation of finite Series. Sigma notation In the last lesson we met the following rules. 1)1 + 2 + 3 + …… + n = (n/2)(n+1) 2)1 2 + 2 2.](https://reader036.fdocuments.net/reader036/viewer/2022071806/56649db35503460f94aa2803/html5/thumbnails/9.jpg)
Question
9920
311561315
)130(2
30)1302)(130(
6
30
)1n(2
n)1n2)(1n(
6
n
rr)1r(r
)13(30...........4)(33)(22)(1 is What30
1
30
1
230
1r
Try this question
![Page 10: Further Pure 1 Summation of finite Series. Sigma notation In the last lesson we met the following rules. 1)1 + 2 + 3 + …… + n = (n/2)(n+1) 2)1 2 + 2 2.](https://reader036.fdocuments.net/reader036/viewer/2022071806/56649db35503460f94aa2803/html5/thumbnails/10.jpg)
Question
9850
)1555(9920
62
5116
6
5 9920
)15(2
5)152)(15(
6
5 9920
)1n(2
n)1n2)(1n(
6
n 9920
rr 9920
)1r(r)1r(r)1r(r
)13(30...........9)(88)(77)(6 is What
5
1
5
1
2
5
1r
30
1r
30
6r
Here the sum starts at r = 6.
This is not as complicated as it may seem.
All you need to do is take of the first 5 terms.
So the sum from 6 to 30 is the sum to 30 minus the sum to 5.
![Page 11: Further Pure 1 Summation of finite Series. Sigma notation In the last lesson we met the following rules. 1)1 + 2 + 3 + …… + n = (n/2)(n+1) 2)1 2 + 2 2.](https://reader036.fdocuments.net/reader036/viewer/2022071806/56649db35503460f94aa2803/html5/thumbnails/11.jpg)
Questions
)2r)(1r(r )d
)2r)(1r( )c
)4r3r4( )b
)4r2r6( )a
n
1r
n
1r
n
1r
3
n
1r
2
Here are some questions for you to find the nth terms of.
The solutions are on the next two slides
![Page 12: Further Pure 1 Summation of finite Series. Sigma notation In the last lesson we met the following rules. 1)1 + 2 + 3 + …… + n = (n/2)(n+1) 2)1 2 + 2 2.](https://reader036.fdocuments.net/reader036/viewer/2022071806/56649db35503460f94aa2803/html5/thumbnails/12.jpg)
Solutions
3)2n2n(n
6n4n2n
4nnnn3n2n
4n1)n(n1)1)(2nn(n
4n2
1)2n(n
6
1)1)(2n6n(n
14r2r6
4r2r6
)4r2r6(
2
23
223
n
1r
n
1r
n
1r
2
n
1r
n
1r
n
1r
2
n
1r
2
)9nn2(n
)83n324nn2(n
)81)(n31)(n2(n
n81)(nn31)(nn2
n41)(n2
3n1)(n
4
4n
14r3r4
4r3r4
)4r3r4(
2
2
2
2
2
n
1r
n
1r
n
1r
3
n
1r
n
1r
n
1r
3
n
1r
3
![Page 13: Further Pure 1 Summation of finite Series. Sigma notation In the last lesson we met the following rules. 1)1 + 2 + 3 + …… + n = (n/2)(n+1) 2)1 2 + 2 2.](https://reader036.fdocuments.net/reader036/viewer/2022071806/56649db35503460f94aa2803/html5/thumbnails/13.jpg)
Solutions
]11n6n[3
n
]22n12n[26
n
]129n91n3n[26
n
]12)1n(9)1n2)(1n[(6
n
n2)1n(2
n3)1n2)(1n(
6
n
12r3r
2r3r
)2r)(1r(
2
2
2
n
1r
n
1r
n
1r
2
n
1r
n
1r
n
1r
2
n
1r
3)2)(n1)(n(n4
n
]6n51)[n(n4
n
]42n4n1)[n(n4
n
]4)1n2(2)1n1)[n((n4
n
)1n(2
n2)1n2)(1n(
6
n3)1n(
4
n
r2r3r
r2r3r
)2r)(1r(r
2
2
22
n
1r
n
1r
2n
1r
3
n
1r
n
1r
2n
1r
3
n
1r
![Page 14: Further Pure 1 Summation of finite Series. Sigma notation In the last lesson we met the following rules. 1)1 + 2 + 3 + …… + n = (n/2)(n+1) 2)1 2 + 2 2.](https://reader036.fdocuments.net/reader036/viewer/2022071806/56649db35503460f94aa2803/html5/thumbnails/14.jpg)
Summation of a finite Series
When Carl Friedrich Gauss was a boy in elementary school his teacher asked his class to add up the first 100 numbers.
S100 = 1 + 2 + 3 + …………… + 100
Gauss had a flash of mathematical genius and realised that the sum had 50 pairs of 101
Therefore S100 = 50 × 101
= 5 050 From this we can come up with the formula for the
sum of the first n numbers. Sn = (n/2)(n+1)
We have met this result a few times already.
![Page 15: Further Pure 1 Summation of finite Series. Sigma notation In the last lesson we met the following rules. 1)1 + 2 + 3 + …… + n = (n/2)(n+1) 2)1 2 + 2 2.](https://reader036.fdocuments.net/reader036/viewer/2022071806/56649db35503460f94aa2803/html5/thumbnails/15.jpg)
Method of differences
We can prove the same result using a different method.
The method of differences.
![Page 16: Further Pure 1 Summation of finite Series. Sigma notation In the last lesson we met the following rules. 1)1 + 2 + 3 + …… + n = (n/2)(n+1) 2)1 2 + 2 2.](https://reader036.fdocuments.net/reader036/viewer/2022071806/56649db35503460f94aa2803/html5/thumbnails/16.jpg)
Use the method of differences to find the sum to 30 of the following example.
Solution to part ii is on the next 2 slides. You covered adding fractions in C2 and should
be able to get the answer.
Example 1
3130
1.......
43
1
32
1
21
1 find Hence )ii
)1r(r
1
1r
1
r
1 that Show )i
![Page 17: Further Pure 1 Summation of finite Series. Sigma notation In the last lesson we met the following rules. 1)1 + 2 + 3 + …… + n = (n/2)(n+1) 2)1 2 + 2 2.](https://reader036.fdocuments.net/reader036/viewer/2022071806/56649db35503460f94aa2803/html5/thumbnails/17.jpg)
Example 1
31
30
31
1 1
31
1
30
1
30
1
29
1
......... 4
1
3
1
3
1
2
1
2
1
1
1
1r
1
r
1
1)r(r
1
3130
1.......
43
1
32
1
21
1 n
1r
30
1r
31
30
1)r(r
130
1r
We can use the identity to re-arrange the question.
Now write the summation out long hand. Starting with r = 1.
Then r = 2,3 etc. Write out the last 2 or 3 terms. Having written out the full summation you
can spot that parts of the sum cancel. The bits that are left do not cancel and we
can sort out the sum.
![Page 18: Further Pure 1 Summation of finite Series. Sigma notation In the last lesson we met the following rules. 1)1 + 2 + 3 + …… + n = (n/2)(n+1) 2)1 2 + 2 2.](https://reader036.fdocuments.net/reader036/viewer/2022071806/56649db35503460f94aa2803/html5/thumbnails/18.jpg)
Example 2
2))(r1r(r
4r find Hence )ii
2))(r1r(r
4r
2r
1
1r
3
r
2 that Show )i
n
1r
Solution to part ii is on the next 2 slides. You covered adding fractions in C2 and
should be able to get the answer.
In this next example we will find the sum to n.
![Page 19: Further Pure 1 Summation of finite Series. Sigma notation In the last lesson we met the following rules. 1)1 + 2 + 3 + …… + n = (n/2)(n+1) 2)1 2 + 2 2.](https://reader036.fdocuments.net/reader036/viewer/2022071806/56649db35503460f94aa2803/html5/thumbnails/19.jpg)
Example 2
2n
1
1n
3
n
2
1n
1
n
3
1-n
2
n
1
1n
3
2-n
2
......... 6
1
5
3
4
2
5
1
4
3
3
2
4
1
3
3
2
2
3
1
2
3
1
2
2r
1
1r
3
r
2
2)1)(rr(r
4r n
1r
n
1r
2n
1
1n
2
2
3
2n
1
1n
3
1n
1
2
2
2
3
1
2
2)1)(rr(r
4rn
1r
We can use the identity to re-arrange the question.
Now write the summation out long hand. Starting with r = 1.
Then r = 2,3 etc. Write out the last 3 terms. Having written out the full summation you
can spot that parts of the sum cancel. The bits that are left do not cancel and we
can sort out the algebra.