1. Sequences and Series (Unit 1C)
Explicit Formula & Summing a Series

2. Arithmetic Sequence Example: 6, 8, 10, 12,…
An arithmetic sequence is a sequence of numbers that has a constant difference between every two consecutive terms.
Example: 6, 8, 10, 12,…
Each term is increasing by 2
Constant difference just means each successive number increases or decreases by the same amount!

3. Arithmetic Series
An arithmetic series is the sum of the terms in an arithmetic sequence.
Ex. 1, 2, 3, 4… (Arithmetic sequence)
… (Arithmetic series)

4. Arithmetic Sequence
To find an exact term in an arithmetic sequence use the ________ formula.
Explicit
an = d(n – 1) + a1
Insert the
common difference
Insert the
1st term

5. Arithmetic Sequence
Find the explicit formula for the following sequence: 5, 11, 17, 23, ... 6 6 6
an = d(n – 1) + a1 5 6
Insert the
common difference
Insert the
1st term

6. Try Again:
Find the explicit formula for the following sequence: 15, 13, 11, 9 ... -2 -2 -2
an = d(n – 1) + a1
= - 83 -2 15 50 50
Now type -2 (50 - 1) +15 = in your calculator
The 50th term in this sequence is -83
Insert the
common difference
Insert the
1st term
Now, suppose I want to know the 50th term in this sequence.
Use the Explicit Formula above and let n = 50.

7. Try Again:
Find the explicit formula for the following sequence: 50, 45.5, 41,
When difficult, Use your calculator to find the difference by
subtracting 2nd term from the
1st term:
45.5 _
-4.5 = d
-4.5
-4.5
-4.5
an = d (n – 1) + a1
-4.5 50
 Using your calculator, Input what you see that comes after the = sign
a25 = -4.5 (25 – 1) + 50
Insert the
common difference
a25 = -58 is the 25th term.
Insert the
1st term
Find the 25th term in this sequence.
Use the Explicit Formula above and replace n with 25.

8. The can pyramid… How many cans are there in this pyramid?
How many cans are there in a pyramid with 100 cans on the bottom row?

9. Back to the pyramid… We wanted to work out the sum of:…
If we write it out in reverse we get…. …
101 +
101 +
101 +…..
101 x 100 = 10100
How many times do we add 101 together?
10100 / 2 = 5050
That’s correct! Half our answer or simply divide by 2!
The summing pattern above is cool, but our answer is now TOO big. What do we need to do to get the correct answer?

10. Let’s Review Steps We wanted to work out the sum of:… …
101 +
101 +
101 +…..
101 x 100 pairs = 10100
10100 / 2 = 5050
That’s correct! Half our answer or simply divide by 2!

11. Activity 1
Work out the sum of the first 50 positive integers ( ) using the same method as used with the pyramid.
Stop Here... And allow students time to work this problem on their paper before moving ahead.

12. Activity 1
Work out the sum of the first 50 positive integers ( ) using the same method as the pyramid.
Check your answer....
Did you get 1275? If so you are CORRECT!

13. Activity 2
Work out the sum of all the ODD numbers from 21 up to 99 using the same method as before.
You MAY need to think a bit harder to determine the total number of pairs since we did not start at 1! Remember to write your list of numbers you are summing forwards and backwards just as you were shown earlier with the pyramid problem! You CAN DO IT !
STOP HERE! Allow students time to work the problem before moving ahead!

14. Find the Sum of all the odd numbers from
Arithmetic Series
Find the Sum of all the odd numbers from
21 up to SOLUTION! … …
__________________________________________________________ …
Did you get there are 40 pairs of numbers? n = 40
SUM = 40(120) = 4800 divided by 2 = 2400
Because we counted every number twice
The SUM of this sequence, S40 = 2400!

15. Arithmetic Series Wah Lah! 21 + 23 + 25 + … + 95 + 97 + 99
a1 = first term
an = last term … …
__________________________________________________________ …
We had 40(120) = 4800 divided by 2 = 2400.
You have uncovered a FORMULA for summing series!
Sum n terms = nbr. of terms (first term + last term)
divided by 2
Sn = n (a1 + an) 2
S40 = 40 ( )  40 (120)  2400
Wah Lah!

16. Student Practice
Find the sum of the following series using the formula
1) Given: …
S1000 = _____
2) Given: …
S600 = _____
Sn = n (a1 + an) 2
STOP HERE! Allow students time to work the problems before moving ahead!

17. Student Practice
Find the sum of the following series using the formula
1) Given: …
S1000 = _____
2) Given: …
S600 = _____
Sn = n (a1 + an) 2
S1000 = ( ) = 500(1001) = 500,500 2
S600 = 600 ( ) = 300(602) = 180,600 2

18. CLICK ON the LINK BELOW AND COMPLETE
THE 1ST PAGE ONLY OF THIS ONLINE WORKSHEET .
REFER BACK TO YOUR PRESENTATION NOTES, IF YOU NEED HELP.
NOTE: YOU CAN CHECK YOUR WORK…ANSWERS CAN BE FOUND ON LAST PAGE OF THIS WORKSHEET.
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