SME
Grade 12 Math · Problem type
The sum of squares
Use this for “Prove 1² + 2² + … + n² = n(n + 1)(2n + 1)/6 by induction.”
Chart
Terms: 1, 4, 9 (the squares n²).
a₃ = 3² = 9
The
partial
sum
adds
the
bars:
S
=
1
+
4
+
9
=
3
×
4
×
7
6
=
14.
This is an example. Type your own values to start a new problem.
1²
+
2²
+
…
+
n
2
=
(
+
1)(2
×
+
1)
6
=
S
a
Next term
worked out (not typed)
T
S plus the next term
worked out (not typed)
F
The formula at n + 1
worked out (not typed)
Formulas
S
=
n
×
(
n
+
1)
×
2 ×
n
+ 1
6
14
=
3
×
(3
+
1)
×
2 × 3 + 1
6
a
=
(
n
+ 1)
2
16
=
(3 + 1)
2
T
=
S
+
a
30 = 14 + 16
F
=
(
n
+
1)
×
(
n
+
2)
×
2 ×
n
+ 3
6
30
=
(3
+
1)
×
(3
+
2)
×
2 × 3 + 3
6
Assumptions
•
Base case: check the formula at n = 1.
•
Step: assume it holds at n = k; adding term k + 1 must give the formula at k + 1.
•
Here: k(k + 1)(2k + 1)/6 + (k + 1)² = (k + 1)(k + 2)(2k + 3)/6; factor out k + 1 to see it.
Step-by-step
WE KNOW
n = 3
FIND
sum of the first n terms (S), next term (a), S plus the next term (T), the formula at n + 1 (F)
Step 1 · Find sum of the first n terms (S)
Use S = n × (n + 1) × (2 × n + 1) ÷ 6
The
formula
at
n
:
1
2
+
2
2
+
…
+
n
2
=
n
(
n
+
1)(
2
n
+
1)/6.
S
=
3
×
(3
+
1)
×
2 × 3 + 1
6
S
=
3
×
4
×
6 + 1
6
S
=
3
×
4
×
7
6
S
=
12
×
7
6
S
=
84
6
S
=
14
Step 2 · Find next term (a)
Use a = (n + 1)²
The
next
term,
number
n
+
1:
(
n
+ 1)
2
.
a
=
(3 + 1)
2
a
=
4
2
a
=
16
Step 3 · Find S plus the next term (T)
Use T = S + a
If
the
formula
holds
at
n
,
the
sum
through
term
n
+
1
is
S
plus
the
next
term.
T
=
14
+
16
T
=
30
Step 4 · Find the formula at n + 1 (F)
Use F = (n + 1) × (n + 2) × (2 × n + 3) ÷ 6
The
formula
with
n
+
1
in
place
of
n
.
F
=
(3
+
1)
×
(3
+
2)
×
2 × 3 + 3
6
F
=
4
×
5
×
6 + 3
6
F
=
4
×
5
×
9
6
F
=
20
×
9
6
F
=
180
6
F
=
30
→
T
=
F
=
30:
adding
the
next
term
gives
the
formula
at
n
+
1.
With
k
for
n
:
k(k
+
1)(2k
+
1)/6
+
(k + 1)
2
=
(k
+
1)(2k²
+
7k
+
6)/6
=
(k
+
1)(k
+
2)(2k
+
3)/6,
since
2k²
+
7k
+
6
=
(k
+
2)(2k
+
3)
Check
Put the numbers back in. Both sides should match.
14
=
3
×
(3
+
1)
×
2 × 3 + 1
6
✓
16
=
(3 + 1)
2
✓
30 = 14 + 16 ✓
30
=
(3
+
1)
×
(3
+
2)
×
2 × 3 + 3
6
✓
Topic check
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