Specialization Examples: 43 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Specialization.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Applying a general theorem or formula to a specific case by substituting particular values for the variables or parameters.

What does this general statement say about MY specific situation?

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Specialization applies a general theorem or formula to a concrete situation by substituting particular values for its variables.

Common stuck point: The procedure for specialization is the easy part; the trap is substituting values into the wrong general formula. Asking "Am I taking a general formula or theorem and feeding it specific values for a single concrete case?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I taking a general formula or theorem and feeding it specific values for a single concrete case?

Worked Examples

Example 1

easy

The Binomial Theorem states

(a+b)^n = \sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k

. Specialise to

a=1, b=1

and

a=1, b=-1

to obtain two identities.

Answer

\sum_{k = 0}^{n} (\binom{n}{k}) = 2^{n}, \sum_{k = 0}^{n} (\binom{n}{k}) (- 1)^{k} = 0

First step

The Binomial Theorem gives

(a+b)^n = \sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k

. Specialization means substituting specific values for the parameters to obtain concrete identities.

Full solution

2
Substitute $a=1, b=1$ : $(1+1)^n = \sum_{k=0}^{n}\binom{n}{k}1^{n-k}\cdot 1^k = \sum_{k=0}^{n}\binom{n}{k}$ . Since the left side equals $2^n$ , we get the identity $\sum_{k=0}^{n}\binom{n}{k} = 2^n$ .
3
Substitute $a=1, b=-1$ : $(1-1)^n = \sum_{k=0}^{n}\binom{n}{k}(-1)^k$ . For $n \ge 1$ the left side is $0^n = 0$ , giving the alternating-sum identity $\sum_{k=0}^{n}\binom{n}{k}(-1)^k = 0$ .

Specialisation plugs specific values into a general formula to obtain particular results. The Binomial Theorem is a powerful source of combinatorial identities via specialisation.

Example 2

medium

The AM-GM inequality states: for positive reals

a,b

\frac{a+b}{2} \ge \sqrt{ab}

. Specialise to

a = x^2

and

b = \frac{1}{x^2}

(for

x \ne 0

) and state what you get.

Example 3

medium

The Pythagorean identity is

\sin^2\theta+\cos^2\theta=1

. Specialize at

\theta=60^\circ

to verify it.

Example 4

medium

Specialize Heron's formula

A=\sqrt{s(s-a)(s-b)(s-c)}

(with

s=\tfrac{a+b+c}{2}

) to a

3

4

5

triangle. Confirm the area.

Example 5

hard

De Moivre:

(\cos\theta+i\sin\theta)^n=\cos(n\theta)+i\sin(n\theta)

. Specialize to

\theta=60^\circ, n=3

and identify the result.

Example 6

medium

Specialize the dot product formula

\mathbf u\cdot\mathbf v=u_1v_1+u_2v_2+u_3v_3

\mathbf u=(1,2,3), \mathbf v=(4,-1,2)

Example 7

challenge

The general formula

\dfrac{1}{ab}=\dfrac{1}{b-a}\left(\dfrac{1}{a}-\dfrac{1}{b}\right)

holds for

a\ne b

. Specialize to

a=k, b=k+1

and use it to telescope

\sum_{k=1}^{n}\dfrac{1}{k(k+1)}

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

The general formula for the sum of a geometric series is

S_n = \frac{a(r^n-1)}{r-1}

. Specialise to

a=1, r=2

and compute

S_5

Example 2

medium

The general derivative rule is

(x^n)' = nx^{n-1}

. Specialise to find the derivatives of

x^3

x^{1/2}

, and

x^{-1}

Example 3

easy

The area of a rectangle is

A = lw

. Specialize to a square of side

s

Example 4

easy

Use

(a+b)^2=a^2+2ab+b^2

to expand

(x+1)^2

Example 5

easy

The general line is

y=mx+b

. Specialize to slope

2

and intercept

-3

Example 6

easy

Specialize

a^n

to find

a^0

for

a \ne 0

Example 7

easy

The quadratic formula gives roots of

ax^2+bx+c=0

. Specialize to

x^2-5x+6=0

Example 8

easy

Specialize the circle area

A=\pi r^2

to radius

3

Example 9

easy

The sum

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

. Specialize to

n=10

Example 10

easy

Specialize the distributive law

a(b+c)=ab+ac

to expand

5(x+2)

Example 11

medium

Specialize the law of cosines

c^2=a^2+b^2-2ab\cos C

C=90^\circ

. What do you recover?

Example 12

medium

Specialize

\binom{n}{k}=\frac{n!}{k!(n-k)!}

to compute

\binom{5}{2}

Example 13

medium

Specialize the geometric series sum

\frac{a(1-r^n)}{1-r}

a=1,r=\frac12,n=3

Example 14

medium

Specialize

f(x)=ax^2+bx+c

to find its value at

x=0

. What does this reveal about

c

Example 15

medium

Specialize the AM-GM inequality

\frac{a+b}{2}\ge\sqrt{ab}

a=b

. What happens?

Example 16

medium

Specialize the dot product formula

\vec{u}\cdot\vec{v}=|\vec{u}||\vec{v}|\cos\theta

to perpendicular vectors. What is the result?

Example 17

challenge

Specialize the binomial theorem

(a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k

a=b=1

. What identity emerges?

Example 18

challenge

Specialize the derivative power rule and chain rule to differentiate

f(x)=(3x)^2

two ways; confirm they agree.

Example 19

challenge

The general solution of

\frac{dy}{dx}=ky

y=Ce^{kx}

. Specialize to

k=1

C=2

, and evaluate at

x=0

Example 20

medium

Specialize the cosine double-angle formula

\cos2\theta=1-2\sin^2\theta

\theta=30^\circ

Example 21

medium

Specialize the slope formula

m=\frac{y_2-y_1}{x_2-x_1}

to the points

(1,2)

and

(4,8)

Example 22

medium

Specialize the compound-interest formula

A=P(1+r)^t

P=100, r=0.1, t=2

Example 23

easy

Specialize the identity

a^2 - b^2 = (a-b)(a+b)

with

a=7, b=3

to compute

7^2 - 3^2

Example 24

easy

Specialize the volume formula

V=\pi r^2 h

for a cylinder with

r=2

and

h=5

Example 25

easy

Specialize the linear function

f(x)=mx+b

m=-1, b=4

. Compute

f(3)

Example 26

easy

Specialize the sine sum identity

\sin(A+B)=\sin A\cos B+\cos A\sin B

A=B=45^\circ

and simplify.

Example 27

medium

Specialize the binomial theorem

(a+b)^n=\sum\binom{n}{k}a^{n-k}b^k

n=3

a=x, b=2

. Write the expanded form.

Example 28

medium

Specialize the matrix-vector product

A\mathbf v

with

A=\begin{pmatrix}1&2\\3&4\end{pmatrix}

and

\mathbf v=\begin{pmatrix}1\\1\end{pmatrix}

Example 29

medium

Specialize the integral

\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C

(for

n\ne -1

) to

n=4

Example 30

medium

Specialize the recurrence

a_{n+1}=2a_n+1

a_0=0

and list

a_1, a_2, a_3

Example 31

hard

Specialize the Taylor series

e^x=\sum_{n=0}^\infty \dfrac{x^n}{n!}

x=1

, then to the first four terms. Report the partial sum.

Example 32

hard

The Cauchy-Schwarz inequality says

(a_1b_1+a_2b_2)^2\le (a_1^2+a_2^2)(b_1^2+b_2^2)

. Specialize to

a_1=a_2=1, b_1=x, b_2=y

and state the resulting inequality.

Example 33

hard

Specialize the law of sines

\dfrac{a}{\sin A}=\dfrac{b}{\sin B}

A=30^\circ, B=45^\circ, a=10

. Find

b

Example 34

medium

Specialize the identity

\log(xy)=\log x+\log y

x=8, y=125

(base 10).

Example 35

hard

Specialize the quadratic formula to

a=2, b=4, c=-30

and find the roots.

Example 36

challenge

Vandermonde's identity:

\sum_{k=0}^{r}\binom{m}{k}\binom{n}{r-k}=\binom{m+n}{r}

. Specialize to

m=n=r

and state the result.

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Related Concepts

Generalization Edge Cases

Background Knowledge

These ideas may be useful before you work through the harder examples.

generalization