Mathematical Induction Worksheet
Fill in the missing details in each of the following proofs by induction.
1. Show that the sequence defined by an = an −1 + 2n −1 for n ≥ 2 , where a1 = 1 , is
equivalently defined by the closed formula an = 2n − 1 .
Proof by induction (i.e., Reader agrees to check in order)
n
(1) Next term from the recursive description
(2) Closed formula 2n − 1
(3) Equal?
1
2
3
4
Let m be the first line in this table not yet checked. Fill in as much of the line before this as possible:
m-1
Now am = ___________________ (from the recurrence relation)
= ___________________ (from looking above in the table)
= ___________________ (algebra)
= ___________________ (algebra)
Hence this line is also true. By the Principle of Mathematical Induction, the closed formula works
for all values of the sequence.
Page 1 of 4
1
1
for n ≥ 2 , where a1 = , is
n(n + 1)
2
n
.
equivalently defined by the closed formula an =
n +1
2. Show that the sequence defined by an = an −1 +
Proof by induction (i.e., Reader agrees to check in order)
n
(1) Next term from the recursive description
(2) Closed formula n
n +1
(3) Equal?
1
2
3
4
Let m be the first line in this table not yet checked. Fill in as much of the line before this as possible:
m-1
Now am = ___________________ (from the recurrence relation)
= ___________________ (from looking above in the table)
= ___________________ (algebra)
= ___________________ (algebra)
Hence this line is also true. By the Principle of Mathematical Induction, the closed formula works
for all values of the sequence.
Page 2 of 4
n
3. Show that
∑ 2k = n
2
+ n for all n ≥ 1.
k =1
Proof by induction (i.e., Reader agrees to check in order)
n
(1) Next sum
(2) Closed formula n2 + n
(3) Equal?
1
2
3
4
Let m be the first line in this table not yet checked. Fill in as much of the line before this as possible:
m-1
m
Now
∑ 2k
= 2 + 4 + … + (2m-2) + 2m
(from the sum notation)
= (__________________) + _____
= ___________________
= ___________________
= ___________________
(algebra)
(from looking above in the table)
(algebra)
(algebra)
k =1
Hence this line is also true. By the Principle of Mathematical Induction, the closed formula works
for all values of the sequence.
Page 3 of 4
n
4. Claim:
∑ (2i − 1) = n
2
for each n ≥ 1 .
i =1
1
Proof by Induction: Since
∑ (2i − 1) = ____ and 1
2
= 1 the claim holds for n=1.
i =1
Let m > 1 be the first value that has not yet been verified. That is, we have already verified
n
that
∑ (2i − 1) = n
2
for 1 ≤ n ≤ m -1 where m -1 is some integer.
i =1
In particular, when n = m − 1 , we therefore know that
m −1
∑ (2i − 1) = ______________
i =1
Next we show that the claim holds when n = m. Therefore,
m
∑ (2i − 1) =
i =1
m −1
∑ (2i − 1) + _________ = _____________________________________
i =1
n
Therefore
∑ (2i − 1) = n
2
for each n ≥ 1 by mathematical induction.
i =1
Use Math Induction to prove the following statement. Be sure to use complete sentences and to fill in all of
the details.
n
5.
Claim:
i =1
Page 4 of 4
1
∑2
i
= 1−
1
for each n ≥ 1 .
2n