Download artihmetic-and-geometric-series-formulas

Sum of an Arithmetic Series: Gauss Trick
Name: _________________________________ A#____
Suppose your teacher asked you to add the numbers from 1 to 100. You would probably begin by adding 1 + 2
+ 3 + ….. + 100, term by term from left to right. Karl Friedrich Gauss (1777- 1855) found another way. Let S
represent the finite series who sum we are trying to find. Since addition is commutative, both equations below
represent this series.
𝑆=
1 + 2 + 3 + ⋯ …. + 98 + 99 + 100
𝑆 = 100 + 99 + 98 + ⋯
3 + 2+ 1
1. Add together the two equations vertically.
2. What do you notice about the sum of each vertically-
𝑆=
+
1 + 2 + 3 + ⋯ …. + 98 + 99 + 100
𝑆 = 100 + 99 + 98 + ⋯ 3 + 2 + 1
aligned pair of quantities on the right side of the
equal sign?
3. How many of such pairs are there?
4. Since each pair has the same sum, use
multiplication to express the sum of all pairs on the
right side.
5. Solve your equation for S. This is the sum of the
integers 1 to 100.
This trick is nice, but will it work when the common difference is not one? Let’s try to generalize Gauss’ trick using a
general arithmetic rule where the first term is 𝑎1 the last term is 𝑎𝑛 and the common difference is d.
6. Fill in the blanks to the right to show the sum of a
general arithmetic sequence of n terms.
𝑆 = 𝑎1 + (𝑎1 + __ ) + (𝑎1 + ___) + ⋯ + (𝑎𝑛 − ___) + 𝑎𝑛
+ 𝑆 = 𝑎𝑛 + (𝑎𝑛 − __) + (𝑎𝑛 − __) + ⋯ + (𝑎1 + __) + 𝑎1
7. Add the equations vertically.
8. What do we notice about the sum of each vertically
aligned pair of quantities on the right had side of the equal
sign?
9. How many pairs are there?
10. Since each pair has the same sum, use multiplication
to express the sum of all pairs on the right side.
11. Solve your equation for S. This is the general formula
for the sum of all finite arithmetic sequences.
12. To compute any finite arithmetic series you add the _____________ term and the ___________term of the series,
multiply the sum by the number of _________ of the series, and divide by ______.
Now practice your formula for a finite arithmetic series by completing PG. 287 #30, 31, 33, 35 [You must show work!]
Sum of a Geometric Series
A#_______
According to legend, the game of chess was invented by a Persian nobleman. The sultan asked the nobleman what he
wanted as a reward. The nobleman took out a board that had 64 squares. He asked the sultan to put 1 grain of wheat
on the first square, 2 grains on the second, 4 on the third, and so on, doubling the number of grains on each successive
square.
1. Complete the table showing the number of grains the nobleman would receive by the time the sultan had reached
the sixth square.
Square Number, n Number of Grains on that Square, an Total Number of Grains on the Board, Sn
1
1
1
2
2
3
3
4
7
4
5
6
2. Write the explicit formula for the number of grains of rice on any square.
3. Use your explicit formula to find 𝑎15 . What does this number represent?
4. Using your explicit formula from #2 find 𝑎𝑛+1 . [Hint: Plug in n+1 for n in your explicit formula and simplify]
5. Compare the number of total grains on the board when 4 squares are covered, 𝑆4 , to the number of grains on square
5, 𝑎5 . What do you notice?
6. Write an equation relating 𝑆𝑛 with 𝑎𝑛+1. Think back to what you saw in #5.
7. Now rewrite your equation from #6 in terms of n by replacing 𝑎𝑛+1 with your result from #4.
8. Your answer in #7 is an explicit formula to find 𝑆𝑛 for this geometric series. Use your formula to find 𝑆29 . What does
this number represent?
9. The general rule for computing a geometric series is 𝑆 =
𝐺1 (𝑟 𝑛 −1)
.
(𝑟−1)
In our rice example above, identify 𝐺1 and r.
10. Does out equation in #7 match the general rule in #9? Why does it look different?
Now practice your formula for a finite geometric series by completing PG. 288 #47, 50, 51, 52 [You must show work!]