SME
Grade 11 Math · Problem type
Sigma notation
Use this for “Evaluate the sum from k = 1 to 8 of (3k − 1).”
Chart
Terms: 2, 5, 8, 11, …, 23 (add d = 3 each time).
a₈ = 2 + 7 × 3 = 23
S₈
=
8
×
2 + 23
2
=
100
This is an example. Type your own values to start a new problem.
c
Coefficient of k
example
e
Constant
example
n
Last value of k
example
a₁
First term (k = 1)
worked out (not typed)
aₙ
Last term (k = n)
worked out (not typed)
S
Sum
worked out (not typed)
Formulas
a₁
=
c
×
1
+
e
2 = 3 × 1 + (−1)
aₙ
=
c
×
n
+
e
23 = 3 × 8 + (−1)
S
=
n
×
a₁
+ aₙ
2
100
=
8
×
2 + 23
2
Assumptions
•
Σ from k = 1 to n of (ck + e) adds the terms for k = 1, 2, …, n.
•
The terms go up by c each time, so the sum is an arithmetic series.
•
Find the first and last terms, then Sₙ = n(a₁ + aₙ) ÷ 2.
Step-by-step
WE KNOW
c = 3, e = −1, n = 8
FIND
first term (k = 1) (a₁), last term (k = n) (aₙ), sum (S)
Step 1 · Find first term (k = 1): a₁
Use a₁ = c × 1 + e
S
=
8
Σ
k
=1
(3
k
− 1)
adds
the
terms
for
k
=
1
to
8.
Put
k
=
1
into
ck
+
e
.
a₁
=
3
×
1
−
1
a₁
=
2
Step 2 · Find last term (k = n): aₙ
Use aₙ = c × n + e
Put
k
=
n
into
ck
+
e
.
aₙ = 3 × 8 − 1
aₙ = 24 − 1
aₙ = 23
Step 3 · Find sum (S)
Use S = n × (a₁ + aₙ) ÷ 2
n
2
pairs
of
first
plus
last,
as
in
any
arithmetic
series.
S
=
8
×
2 + 23
2
S
=
8
×
25
2
S
=
200
2
S
=
100
Check
Put the numbers back in. Both sides should match.
2 = 3 × 1 − 1 ✓
23 = 3 × 8 − 1 ✓
8
Σ
k
=1
(3
k
− 1)
=
100
✓
Topic check
See what you know: 5 questions on this topic
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