📌 Example 1.
Express the geometric series, 7+14+28+56+112+224, using summation notation.
✍ Solution:
Write a formula: a_n=a₁⋅r^(n-1)
List the information: a₁=7

r=14/7
n=6 (use 6 on the Σ symbol, but leave it unknown in the argument.)
a_n=?
Substitute and solve: a_n=7⋅2^(n-1).
Apply the Sigma notation: ∑⁶_n=1 (7⋅2^(n-1) )
Therefore, 7+14+28+56+112+224=∑⁶_n=1 (7⋅2^(n-1) ) .

📌 Ex2. Write each geometric series in sigma notation.
📌 Ex2a. 3+12+48+…+3072.
✍ Solution:
Find the common ratio.

12÷3=4
48÷12=4
Next, determine the upper bound.
a₄=48⋅4=192
a₅=192⋅4=768
a₆=768⋅4=3072
Write an explicit formula for the sequence.
a_n=a₁⋅r^(n-1)
=3⋅4^(n-1)
Therefore, in sigma notation the series 3+12+48+…+3072 can be written as ∑⁶_n=1 3⋅4^(n-1).

📌 Ex2b. 9+18+36+…+1152.
✍ Solution:
Find the common ratio.

18÷9=2
36÷18=2

Next, determine the upper bound.

a₄=36⋅2=72
a₅=72⋅2=144
a₆=144⋅2=288
a₇=288⋅2=576
a₈=576⋅2=1152

Write an explicit formula for the sequence.

a_n=a₁⋅r^(n-1)
=9⋅2^(n-1)
Therefore, in sigma notation the series 9+18+36+…+1152 can be written as ∑⁸_n=1 9⋅2^(n-1).

📌 Ex2c. 50+85+144.5+…+417.605.
✍ Solution:
Find the common ratio.

85÷50=1.7
144.5÷85=1.7

Next, determine the upper bound.

a₄=144.5⋅1.7=245.65
a₅=245.65⋅1.7=417.605

Write an explicit formula for the sequence.

a_n=a₁⋅r^(n-1)
=50⋅(1.7)^(n-1)
Therefore, in sigma notation the series 50+85+144.5+…+417.605 can be written as ∑⁵_n=1 50⋅(1.7)^(n-1).

📌 Ex2d. ⅛-¼+½-…+8.
✍ Solution:
Find the common ratio.

-¼÷⅛=-2
½÷(-¼)=-2

Next, determine the upper bound.

a₄=½⋅(-2)=-1
a₅=-1⋅(-2)=2
a₆=2⋅(-2)=-4
a₇=-4⋅(-2)=8

Write an explicit formula for the sequence.

a_n=a₁⋅r^(n-1)
=⅛⋅(-2)^(n-1)

Therefore, in sigma notation the series ⅛+(-¼)+½+…+8 can be written as ∑⁷_n=1 ⅛⋅(-2)^(n-1).

📌 Ex2e. 0.2-1+5-…-625.
✍ Solution:
Find the common ratio.

-1÷0.2=-5
5÷-1=-5

Next, determine the upper bound.

a₄=5⋅(-5)=-25
a₅=-25⋅(-5)=125
a₆=125⋅(-5)=-625

Write an explicit formula for the sequence.

a_n=a₁⋅r^(n-1)
=0.2⋅(-5)^(n-1)

Therefore, in sigma notation the series 0.2+(-1)+5+…+(-625) can be written as ∑⁶_n=1 0.2⋅(-5)^(n-1).