3.3 Mathematical Induction & the Binomial Theorem

Mathematical Induction & the Binomial Theorem

Mathematical induction and the binomial theorem provide foundational tools for algebraic proof, power expansion, and functional approximations. On the ECAT, proficiency with the general term formula $T_{r+1}$, middle term determination, independent term isolation, and coefficient summation shortcuts is essential for scoring highly.

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1. Principle of Mathematical Induction (PMI)

The Principle of Mathematical Induction is an axiomatic method of proof used to establish that a statement $P(n)$ is true for all natural numbers $n \in \mathbb{N}$ (or for all integers $n \ge n_0$).

The Three Formal Steps

1. Base Step (Condition I): Show that the statement is true for the starting index, typically $n = 1$. That is, establish $P(1)$ as true.
2. Inductive Hypothesis: Assume that the statement is true for an arbitrary positive integer $k$. That is, hypothesize that $P(k)$ holds.
3. Inductive Step (Condition II): Using the inductive hypothesis $P(k)$, prove rigorously that the statement must also be true for the next integer $k+1$. That is, prove $P(k) \implies P(k+1)$.

Conclusion: If both conditions are satisfied, then by PMI, $P(n)$ is true for all integers $n \ge 1$.

Typical Mathematical Applications

• Summation Formulas: Proving standard power identities such as $\sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}$.
• Divisibility Assertions: Proving that an expression is divisible by an integer for all $n$. For example, proving that $a^n - b^n$ is divisible by $a - b$, or that $5^{2n} - 1$ is divisible by 24 for all $n \ge 1$.
• Inequalities: Establishing bounding relations such as $2^n > n^2$ for all integers $n \ge 5$, or Bernoulli's inequality $(1+x)^n \ge 1 + nx$ for $x > -1$ and $n \in \mathbb{N}$.

Test-Day Strategy: ECAT multiple-choice questions rarely require writing out an entire induction proof. Instead, problems usually ask: "For what positive integers is $P(n)$ true?" Candidates should test small values directly ($n = 1, 2, 3$) to quickly eliminate incorrect answer options in seconds.

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2. Binomial Theorem for Positive Integral Exponents ($n \in \mathbb{N}$)

For any real or complex numbers $a$ and $b$, and any positive integer $n$:

$(a+b)^n = {}^n C_0 a^n b^0 + {}^n C_1 a^{n-1} b^1 + {}^n C_2 a^{n-2} b^2 + \dots + {}^n C_r a^{n-r} b^r + \dots + {}^n C_n a^0 b^n$

In compact sigma notation:

$(a+b)^n = \sum_{r=0}^{n} {}^n C_r a^{n-r} b^r$

Fundamental Structural Properties

1. Total Number of Terms: The expansion contains exactly $n + 1$ terms.
2. Degree Homogeneity: In every term $a^{n-r} b^r$, the sum of the exponents of $a$ and $b$ is $(n-r) + r = n$.
3. Symmetry of Coefficients: Because ${}^n C_r = {}^n C_{n-r}$, coefficients equidistant from the beginning and end of the expansion are identical.
4. Sign Alternation: For $(a - b)^n$, terms alternate in sign: $(a-b)^n = \sum_{r=0}^{n} (-1)^r {}^n C_r a^{n-r} b^r$

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3. The General Term ($T_{r+1}$)

Because the expansion index $r$ begins at 0, the $(r+1)$-th term of $(a+b)^n$ is denoted as $T_{r+1}$:

$T_{r+1} = {}^n C_r a^{n-r} b^r$

Important Index Rule: The $k$-th term of the expansion corresponds to setting $r = k - 1$.

• 1st term ($k=1$): $r = 0 \implies T_1 = {}^n C_0 a^n$
• 5th term ($k=5$): $r = 4 \implies T_5 = {}^n C_4 a^{n-4} b^4$
• Last term ($k=n+1$): $r = n \implies T_{n+1} = {}^n C_n b^n$

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4. Middle Terms in the Expansion

The position of the middle term depends entirely on whether the exponent $n$ is even or odd:

Case 1: When $n$ is Even

• The total number of terms is $n + 1$, which is an odd integer.
• Therefore, there is exactly one middle term.
• Its position is the $\left(\frac{n}{2} + 1\right)$-th term (corresponding to index $r = \frac{n}{2}$):

$T_{\text{middle}} = T_{\frac{n}{2}+1} = {}^n C_{n/2} a^{n/2} b^{n/2}$

Case 2: When $n$ is Odd

• The total number of terms is $n + 1$, which is an even integer.
• Therefore, there are two distinct middle terms.
• Their positions are the $\left(\frac{n+1}{2}\right)$-th term and the $\left(\frac{n+3}{2}\right)$-th term:

$T_{\text{middle, 1}} = T_{\frac{n-1}{2}+1} = {}^n C_{\frac{n-1}{2}} a^{\frac{n+1}{2}} b^{\frac{n-1}{2}}$

$T_{\text{middle, 2}} = T_{\frac{n+1}{2}+1} = {}^n C_{\frac{n+1}{2}} a^{\frac{n-1}{2}} b^{\frac{n+1}{2}}$

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5. Term Independent of $x$ (Constant Term)

In expansions involving powers of a variable $x$, such as $\left(A x^p + \frac{B}{x^q}\right)^n$, questions frequently ask for the term independent of $x$ (where the power of $x$ is 0).

Systematic Solution Algorithm

1. Write the general term $T_{r+1}$: $T_{r+1} = {}^n C_r (A x^p)^{n-r} \left(B x^{-q}\right)^r = {}^n C_r A^{n-r} B^r x^{p(n-r) - qr}$
2. Collect all exponents of $x$ into a single expression: $E(r) = p(n-r) - qr = pn - (p+q)r$
3. Set the net exponent equal to zero and solve for $r$: $pn - (p+q)r = 0 \implies r = \frac{pn}{p+q}$
4. Validity Check: The term independent of $x$ exists if and only if $r$ is an integer in the interval $0 \le r \le n$. If $r$ is fractional or negative, no constant term exists in the expansion (its coefficient is 0).
5. Substitute the resolved value of $r$ back into the numerical coefficient of $T_{r+1}$.

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6. Binomial Coefficient Identities

Let $C_r$ denote ${}^n C_r = \binom{n}{r}$. Substituting specific values for $a$ and $b$ into $(a+b)^n$ produces classic coefficient summation identities:

1. Sum of All Binomial Coefficients

Substitute $a = 1$ and $b = 1$ into $(a+b)^n$:

$C_0 + C_1 + C_2 + \dots + C_n = (1+1)^n = 2^n$

2. Alternating Sum of Binomial Coefficients

Substitute $a = 1$ and $b = -1$ into $(a+b)^n$:

$C_0 - C_1 + C_2 - C_3 + \dots + (-1)^n C_n = (1-1)^n = 0$

3. Sum of Even and Odd Coefficients

From the two equations above, adding and subtracting yields:

$C_0 + C_2 + C_4 + \dots = C_1 + C_3 + C_5 + \dots = 2^{n-1}$

4. Advanced Coefficient Summations

• First derivative identity: $\sum_{r=1}^n r C_r = 1 C_1 + 2 C_2 + 3 C_3 + \dots + n C_n = n 2^{n-1}$
• Sum of squares of coefficients: $\sum_{r=0}^n C_r^2 = C_0^2 + C_1^2 + C_2^2 + \dots + C_n^2 = {}^{2n} C_n = \frac{(2n)!}{(n!)^2}$

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7. Binomial Theorem for Any Rational Index ($n \in \mathbb{Q}$)

When the exponent $n$ is a negative integer or a non-integer fraction ($n \notin \mathbb{N}$), the expansion of $(1+x)^n$ requires a modified formulation.

Infinite Series Formulation

For any rational number $n$ and any real $x$ satisfying $|x| < 1$:

$(1+x)^n = 1 + nx + \frac{n(n-1)}{2!} x^2 + \frac{n(n-1)(n-2)}{3!} x^3 + \dots + \frac{n(n-1)\dots(n-r+1)}{r!} x^r + \dots$

Fundamental Differences from the Positive Integral Theorem:

1. Infinite Number of Terms: Because $n$ is not a positive integer, the factor $(n - r + 1)$ never equals zero. The series contains infinitely many terms.
2. Strict Convergence Constraint: The series converges if and only if $|x| < 1$ (i.e., $-1 < x < 1$). If $|x| \ge 1$, the series diverges and cannot be evaluated.
3. Base Normalization Requirement: To expand an expression such as $(a + b)^n$ where $n \notin \mathbb{N}$, factor out $a^n$ to form $a^n \left(1 + \frac{b}{a}\right)^n$, requiring $\left|\frac{b}{a}\right| < 1$.

Canonical Infinite Series to Memorize:

• $(1-x)^{-1} = 1 + x + x^2 + x^3 + \dots \quad (|x| < 1)$
• $(1+x)^{-1} = 1 - x + x^2 - x^3 + \dots \quad (|x| < 1)$
• $(1-x)^{-2} = 1 + 2x + 3x^2 + 4x^3 + \dots \quad (|x| < 1)$
• $(1+x)^{-2} = 1 - 2x + 3x^2 - 4x^3 + \dots \quad (|x| < 1)$
• $(1+x)^{1/2} = 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3 - \dots \quad (|x| < 1)$

Rapid Decimal Approximation Shortcut

When $x$ is very small compared to 1 ($|x| \ll 1$), higher powers of $x$ become negligible, allowing the linear approximation:

$(1+x)^n \approx 1 + nx$

Example: Approximate $\sqrt{1.04} = (1 + 0.04)^{1/2}$:

$\sqrt{1.04} \approx 1 + \frac{1}{2}(0.04) = 1 + 0.02 = 1.02$

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8. Summary Comparison: Positive Integral vs. Rational Index