W16D2 Geometric Series
Warm Up
P
1. Sum 5, 11, 17.... 95
n=8
=800
2. How many terms are in the sequence 2, 6, 18.... 162 ? n = 5
Lesson 36 Geometric Series
Intro Problem: You open a savings account with $300. You earn 5% interest each year. What kind of sequence does this
represent?
Write the rule and find the amount of money you will earn in the 8th year. NOT YOUR TOTAL an = 300(.05)8
You throw a SuperBall on the cement as hard as you can and watch it bounce until it stops. You notice the first bounce reaches
a height of 200ft, but the second bounce reaches only half of that height. How high will the 7th bounce reach? How far (total
distance) has the ball traveled before the 8th bounce?
a7 = 3.125f t
an = 200( 12 )n−1 )
Sum = 396.875 ft
EX 1:
2, 6, 18, 54, 162 242
EX 2:
2, 10, 50... 156,250 39062
Proof for the formula
S = a1 + a2 + a3 + a4 + a5 + a6 + a7
S = a1 + a1 r + a1 r2 + a1 r3 + a1 r4 + a1 r5 + a1 r6
rS = a1 r + a1 r2 + a1 r3 + a1 r4 + a1 r5 + a1 r6 + a1 r7
Subtract
S − rS = a1 + a1 r7
Solve for S
S(1 − r) = a1 + a1 r7
S=
= a1 (1 − r7 )
1−r
Formula for nGeometric Sum:
(1−r )
S = =a11−r
EX 3:
3
, 3, 15....1875
5
EX 4:
64, −32, 16, ....
EX 5:
5, 15, 45....135
EX 6:
1, 2, 4....with 9 terms
10
90, 30, 10...
81
1, 2, 4....with 11 terms
1 3 9
, , sums to 14762. How many terms?
2 2 2
3
3
− 6, 3, − ... −
2
64
4, 12, 36....Sums to 1456. What is an ?
EX 7:
EX 8:
EX 9:
EX 10:
EX 11:
EX 12:
1, −3, 9... − 2187
EX 13:
2, −2, 2....with 14 terms
Exit Pass
1. Find the Sum of W10D4
2. Find the sum of 9, 16, 23...51
n=7
1
16
P7
n=1 =
210
n=6
2343.6
n = 11
42.6875
n=4
200
an = 256
511
n=7
an = 1024
an = 9841.5
n=8
134.9383
2047
n = 10
−11.9531
n=6
an = 972
n=8
S = −1640
an = −2
S=0