SME
Grade 12 Math · Problem type
The sum of odd numbers
Use this for “Prove 1 + 3 + 5 + … + (2n − 1) = n² by induction.”
Chart
Terms: 1, 3, 5, 7, 9 (add d = 2 each time).
a₅ = 1 + 4 × 2 = 9
S₅
=
5
×
1 + 9
2
=
25
This is an example. Type your own values to start a new problem.
1
+
3
+
…
+
(2
×
n
−
1)
=
n
2
=
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
2
25
=
5
2
a
=
2
×
n
+
1
11 = 2 × 5 + 1
T
=
S
+
a
36 = 25 + 11
F
=
(
n
+ 1)
2
36
=
(5 + 1)
2
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: k² + (2k + 1) = (k + 1)², so the step holds for every k.
Step-by-step
WE KNOW
n = 5
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 = n²
The
formula
at
n
:
1
+
3
+
…
+
(
2
n
−
1)
=
n
2
.
S
=
5
2
S
=
25
Step 2 · Find next term (a)
Use a = 2 × n + 1
The
next
term,
number
n
+
1:
2(
n
+
1)
−
1
=
2
n
+
1.
a
=
2
×
5
+
1
a
=
10
+
1
a
=
11
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
=
25
+
11
T
=
36
Step 4 · Find the formula at n + 1 (F)
Use F = (n + 1)²
The
formula
with
n
+
1
in
place
of
n
.
F
=
(5 + 1)
2
F
=
6
2
F
=
36
→
T
=
F
=
36:
adding
the
next
term
gives
the
formula
at
n
+
1.
With
k
for
n
:
k
2
+
(2k
+
1)
=
(k + 1)
2
Check
Put the numbers back in. Both sides should match.
25
=
5
2
✓
11 = 2 × 5 + 1 ✓
36 = 25 + 11 ✓
36
=
(5 + 1)
2
✓
Topic check
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Mathematical induction
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