AlgebraHigh Yield (2 Qs in JEE)

Binomial Theorem Formulas

General term T_{r+1}, middle term, greatest term, binomial coefficients identities, and remainder theorem.

8 Formulas•443+ Questions in Bank

Core Formulas & Mathematical Identities

General Term in Binomial Expansion

(x + y)^n = \sum_{r=0}^n {}^nC_r x^{n-r} y^r, \quad T_{r+1} = {}^nC_r x^{n-r} y^r

Expansion has (n + 1) terms.

Examiner Tip: Independent of x term occurs when power of x in T_{r+1} equals 0.

Binomial Coefficient Sum Identities

\sum_{r=0}^n {}^nC_r = 2^n, \quad \sum_{r=0}^n (-1)^r {}^nC_r = 0, \quad \sum_{r=1}^n r \cdot {}^nC_r = n \cdot 2^{n-1}

Classic combinatorial summation identities.

Examiner Tip: Differentiate (1 + x)^n to find sums with r in the coefficient.

Middle Term in Expansion

n \text{ even} \implies T_{\frac{n}{2} + 1}; \quad n \text{ odd} \implies T_{\frac{n+1}{2}} \text{ and } T_{\frac{n+3}{2}}

Has the greatest binomial coefficient ^nC_{max}.

Examiner Tip: For even n, single middle term has maximum coefficient ^nC_{n/2}.

Fractional / Negative Index Binomial Series

(1 + x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \cdots \quad (|x| < 1), \quad T_{r+1} = \frac{n(n-1)\cdots(n-r+1)}{r!}x^r

Infinite convergent series for any rational n (negative or fractional). General term uses falling factorial in numerator.

Examiner Tip: Common cases: (1+x)^{-1} = 1 - x + x² - x³ + ⋯, (1-x)^{-1} = 1 + x + x² + x³ + ⋯, (1+x)^{-2} = 1 - 2x + 3x² - 4x³ + ⋯. All valid for |x| < 1.

Weighted Binomial Coefficient Sums

\sum_{r=1}^n r \cdot {}^nC_r = n \cdot 2^{n-1}, \quad \sum_{r=1}^n r^2 \cdot {}^nC_r = n(n+1) \cdot 2^{n-2}, \quad \sum_{r=0}^n \frac{1}{r+1}{}^nC_r = \frac{2^{n+1} - 1}{n + 1}

Weighted sums obtained by differentiating or integrating (1+x)^n and substituting x = 1.

Examiner Tip: Differentiate (1+x)^n once → Σ r·C(n,r)·x^(r-1); differentiate again → Σ r(r-1)·C(n,r)·x^(r-2). Add to get r²·C(n,r) sum.

Multinomial Theorem & Number of Terms

(x_1 + x_2 + \dots + x_k)^n = \sum \frac{n!}{r_1! r_2! \dots r_k!} x_1^{r_1} x_2^{r_2} \dots x_k^{r_k}, \quad \text{Total Terms} = {}^{n+k-1}C_{k-1}

Expansion of sum of k variables raised to integer power n where r₁ + r₂ + ... + r_k = n.

Examiner Tip: Coefficient of x₁^{r₁} x₂^{r₂}...x_k^{r_k} is the multinomial coefficient n! / (r₁! r₂!...r_k!).

Numerically Greatest Term (NGT)

\left| \frac{T_{r+1}}{T_r} \right| = \frac{n - r + 1}{r} \left| \frac{y}{x} \right| \ge 1 \implies r \le \frac{n + 1}{1 + |x/y|}

Determines the term having the largest magnitude in the expansion of (x + y)^n.

Examiner Tip: If m = (n+1)/(1 + |x/y|) is an integer, both T_m and T_{m+1} are equal and numerically greatest.

Divisibility & Remainder Finding

(1 + k)^n = 1 + nk + {^nC_2} k^2 + \dots \equiv 1 + nk \pmod{k^2}

Express base as (multiple of divisor ± remainder) to find remainders of large powers.

Examiner Tip: For finding remainder of 7¹⁰³ divided by 25: rewrite as 7 · (49)⁵¹ = 7 · (50 - 1)⁵¹.

Top Examiner Traps & Pitfalls

Frequently tested misconceptions in Binomial Theorem

•The (r+1)-th term has coefficient ^nC_r, NOT ^nC_{r+1}.
•General index binomial series (1+x)^n requires |x| < 1 when n is negative or fractional.

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