Using Sigma Notation to represent Finite Geometric Series

π Example 1.
Express the geometric series, 7+14+28+56+112+224, using summation notation.
β Solution:
Write a formula: an=a1βr(n-1)
List the information: a1=7

r=14/7
n=6 (use 6 on the Ξ£ symbol, but leave it unknown in the argument.)
an=?

Substitute and solve: an=7β2(n-1).
Apply the Sigma notation: β6n=1 (7β2(n-1) )
Therefore, 7+14+28+56+112+224=β6n=1 (7β2(n-1) ) .

π Ex2. Write each geometric series in sigma notation.
π Ex2a. 3+12+48+β¦+3072.
β Solution:
Find the common ratio.

12Γ·3=4
48Γ·12=4

Next, determine the upper bound.
a4=48β4=192
a5=192β4=768
a6=768β4=3072

Write an explicit formula for the sequence.
an=a1βr(n-1)
=3β4(n-1)

Therefore, in sigma notation the series 3+12+48+β¦+3072 can be written as β6n=1 3β4(n-1).

π Ex2b. 9+18+36+β¦+1152.
β Solution:
Find the common ratio.

18Γ·9=2
36Γ·18=2

Next, determine the upper bound.

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

Write an explicit formula for the sequence.

an=a1βr(n-1)
=9β2(n-1)

Therefore, in sigma notation the series 9+18+36+β¦+1152 can be written as β8n=1 9β2(n-1).

π Ex2c. 50+85+144.5+β¦+417.605.
β Solution:
Find the common ratio.

85Γ·50=1.7
144.5Γ·85=1.7

Next, determine the upper bound.

a4=144.5β1.7=245.65
a5=245.65β1.7=417.605

Write an explicit formula for the sequence.

an=a1βr(n-1)
=50β(1.7)(n-1)

Therefore, in sigma notation the series 50+85+144.5+β¦+417.605 can be written as β5n=1 50β(1.7)(n-1).

π Ex2d. β-ΒΌ+Β½-β¦+8.
β Solution:
Find the common ratio.

-ΒΌΓ·β=-2
Β½Γ·(-ΒΌ)=-2

Next, determine the upper bound.

a4=Β½β(-2)=-1
a5=-1β(-2)=2
a6=2β(-2)=-4
a7=-4β(-2)=8

Write an explicit formula for the sequence.

an=a1βr(n-1)
=ββ(-2)(n-1)

Therefore, in sigma notation the series β+(-ΒΌ)+Β½+β¦+8 can be written as β7n=1 ββ(-2)(n-1).

π Ex2e. 0.2-1+5-β¦-625.
β Solution:
Find the common ratio.

-1Γ·0.2=-5
5Γ·-1=-5

Next, determine the upper bound.

a4=5β(-5)=-25
a5=-25β(-5)=125
a6=125β(-5)=-625

Write an explicit formula for the sequence.

an=a1βr(n-1)
=0.2β(-5)(n-1)

Therefore, in sigma notation the series 0.2+(-1)+5+β¦+(-625) can be written as β6n=1 0.2β(-5)(n-1).