Practice Generalization in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

The process of extending a specific result or pattern to hold for a broader class of objects or situations.

Does this pattern work more generally? Can we remove restrictions?

Showing a random 20 of 50 problems.

Example 1

medium

5^0 = 1

and

7^0 = 1

. Generalize

a^0

and state the exception.

Example 2

easy

Specific:

0 + 7 = 7

0 + 11 = 11

. Generalize: what is

0 + n

Example 3

easy

2^3 \cdot 2^4 = 2^7

and

5^2 \cdot 5^6 = 5^8

. Generalize the exponent rule.

Example 4

easy

A right triangle has legs

3

and

4

with hypotenuse

5

. Generalize to legs

a

and

b

with hypotenuse

c

Example 5

medium

(x-1)(x+1) = x^2 - 1

. Generalize to a difference of two squares.

Example 6

medium

2\cdot3=6

shares no structure issue, but generalize: for primes

p

, is

p^2

ever even? Decide and generalize.

Example 7

medium

The formula

1+2+\cdots+n = \frac{n(n+1)}{2}

holds for

n=1,2,3

. State how you would generalise this claim to all positive integers and what technique would be used.

Example 8

hard

\binom{4}{2} = \binom{4}{2}

(trivial). Generalize the symmetry of binomials.

Example 9

easy

From

\frac{1}{2}+\frac{1}{2}=1

, generalize: what is

\frac{1}{n}

added to itself

n

times?

Example 10

medium

\gcd(6,4) = 2

\gcd(15,10) = 5

\gcd(8, 12) = 4

. Generalize: what is

\gcd(a, b)

in terms of common factors?

Example 11

easy

Specific:

3 \times 5 = 15

(odd

\times

odd = odd). Generalise: prove that the product of any two odd integers is odd.

Example 12

medium

\sin 30° = 1/2

\sin 150° = 1/2

. Generalize: when does

\sin\theta = 1/2

Example 13

hard

1 + r + r^2 = \frac{1 - r^3}{1 - r}

for

r \ne 1

. Generalize the finite geometric sum.

Example 14

medium

1\cdot2=2

2\cdot3=6

3\cdot4=12

. Generalize the product of two consecutive integers

n(n+1)

and state its parity.

Example 15

hard

\sum_{k=0}^{n}\binom{n}{k} = 2^n

holds for small

n

. Justify the generalization combinatorially.

Example 16

easy

Specific:

5 - 5 = 0

7 - 7 = 0

. Generalize to any real

a

Example 17

medium

\binom{4}{2} = 6

\binom{5}{2} = 10

\binom{6}{2} = 15

. Generalize

\binom{n}{2}

Example 18

easy

The sum of interior angles is

180^\circ

for a triangle and

360^\circ

for a quadrilateral. Generalize to an

n

-gon.

Example 19

easy

Specific:

3 + (4 + 5) = (3 + 4) + 5

. Generalize this to any

a, b, c

Example 20

medium

1^2 = 1

1^2 + 2^2 = 5

1^2 + 2^2 + 3^2 = 14

. Find a closed formula for

\sum_{k=1}^{n} k^2

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Related Concepts

Abstraction Specialization