SME
Grade 11 Math · Algebra 2
Series and sigma notation (arithmetic and geometric)
Use this for “Find the sum of the first 6 terms of 3, 6, 12, ….”
Chart
Terms: 3, 6, 12, 24, 48, 96 (multiply by r = 2 each time).
a₆ = 3 × 2⁵ = 96
S₆
=
3
×
1 −
2
6
1 − 2
=
189
This is an example. Type your own values to start a new problem.
Sₙ
=
a₁
×
1
−
r
1
−
aₙ
Last term
calculated
Formulas
aₙ
=
a₁
×
r
n
− 1
96
=
3
×
2
6 − 1
Sₙ
=
a₁
×
1 −
r
n
1 −
r
189
=
3
×
1 −
2
6
1 − 2
Assumptions
•
Each term is the one before times the common ratio r.
•
Multiply S by r and subtract: all but two terms cancel, leaving S(1 − r) = a₁(1 − rⁿ).
•
For r = 1 every term is a₁, so S is just n × a₁.
Step-by-step
WE KNOW
n = 6, a₁ = 3, r = 2
FIND
last term (aₙ), sum of the n terms (Sₙ)
Step 1 · Find last term (aₙ)
Use aₙ = a₁ × rⁿ⁻¹
From
the
first
term,
multiply
by
r
,
n
−
1
times.
aₙ
=
3
×
2
6 − 1
aₙ
=
3
×
2
5
aₙ = 3 × 32
aₙ = 96
Step 2 · Find sum of the n terms (Sₙ)
Use Sₙ = a₁ × (1 − rⁿ) ÷ (1 − r)
The
sum
of
a
geometric
series:
a₁
times
(1
−
rⁿ)
over
(1
−
r
).
Sₙ
=
3
×
1 −
2
6
1 − 2
Sₙ
=
3
×
1 − 64
1 − 2
Sₙ
=
3
×
−63
−1
Sₙ = 189
Check
Put the numbers back in. Both sides should match.
96
=
3
×
2
6 − 1
✓
189
=
3
×
1 −
2
6
1 − 2
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.