For some n ≠ 10$n \neq 10$, let the coefficients of the 5th$5^{\text{th}}$, 6th$6^{\text{th}}$ and 7th$7^{\text{th}}$ terms in the binomial expansion of (1 + x)ⁿ⁺⁴$(1 + x)^{n+4}$ be in A.P. Then the largest coefficient in the expansion of (1 + x)ⁿ⁺⁴$(1 + x)^{n+4}$ is :

Related Formula

Algebraic factorizations:

Core Logic

Simplify inside the parentheses before using the Binomial general term formula.

Step 1: Simplify Bracket Terms

First fraction:

Second fraction:

Subtract the two simplified results:

Step 2: Apply Binomial Theorem

The expression simplifies to (x1/3 - x-1/2)¹⁰$\left(x^{1/3} - x^{-1/2}\right)^{10}$. Write the general term Tᵣ₊₁$T_{r+1}$:

Step 3: Solve for Independent Term

Set the net power of x$x$ to zero:

Substitute r=4$r=4$ into the general term expression:

Pattern Recognition

The initial fractions look complex but contain simple hidden identities (a³+b³$a^3+b^3$ and a²-b²$a^2-b^2$). Identifying these transforms a daunting fraction expansion into a classic two-term independent coefficient puzzle.

Chapter Mix

Class 11 Mathematics: Binomial Theorem

Related Formula

For trinomial expansion:

Symmetric coefficients can be extracted by substituting x=1$x=1$ and x=-1$x=-1$.

Core Logic

Let's perform the substitution:

• For x = 1$x = 1$:

• For x = -1$x = -1$:

Step 1: Extracting odd term sum

Subtracting (2) from (1):

Since 3⁵ = 243 3¹⁰ = 59049$3^5 = 243 \implies 3^{10} = 59049$:

Step 2: Finding a₂$a_2$ and k$k$

Using expansion formula to find a₂$a_2$ (coefficient of x²$x^2$):

For q + 2r = 2$q + 2r = 2$:

• Case 1: r=1, q=0 p=9$r=1, q=0 \implies p=9$. Coefficient = (10!)/(9! 0! 1!) = 10$\frac{10!}{9! 0! 1!} = 10$
• Case 2: r=0, q=2 p=8$r=0, q=2 \implies p=8$. Coefficient = (10!)/(8! 2! 0!) = 45$\frac{10!}{8! 2! 0!} = 45$

Now substitute into the target relation:

Pattern Recognition

Substituting standard complex roots or positive units is the fastest way to resolve sum of subset coefficients in polynomial expansions. Always use partition calculations for the early index terms (a₁, a₂$a_1, a_2$) rather than standard multinomial permutations to avoid calculation mistakes.

Chapter Mix

Class 11 Mathematics: Binomial Theorem

Related Formula

Binomial expansion for checking remainders:

Core Logic

Let the target expression be N = ((64)⁶⁴)⁶⁴$N = \left((64)^{64}\right)^{64}$. Using power rules, N = 6464 × 64 = 6464²$N = 64^{64 \times 64} = 64^{64^2}$.

Let the large exponent exponent be n = 64²$n = 64^2$. We need to find the remainder of 64ⁿ$64^n$ when divided by 7$7$.

Step 1: Express Base in Terms of Modulus

Observe that 64 = 63 + 1 = 7 × 9 + 1$64 = 63 + 1 = 7 \times 9 + 1$. Substituting this into the expression: N = (1 + 63)ⁿ$N = (1 + 63)^n$

Expanding using the binomial theorem:

Since 7(9λ)$7(9\lambda)$ is perfectly divisible by 7$7$, the remaining term is 1$1$.

Pattern Recognition

Whenever the base can be written as km + 1$km + 1$, where m$m$ is the divisor, the value of (km + 1)ⁿ ≡ 1ⁿ ≡ 1 m$(km + 1)^n \equiv 1^n \equiv 1 \pmod m$ instantly, regardless of the size or complexity of the exponent layout.

Chapter Mix

Class 11 Mathematics: Binomial Theorem

Related Formula

Core Logic

Formulate general progression steps. Isolate divisor common multiples to find non fraction power configurations tracking perfectly over boundaries.

Step 1: State the General Binomial Term

Step 2: Apply Rational Power Constraints

For expressions to map purely to integer categories, indexing tracker index paths r$r$ must explicitly scale as multiples of 8 over the domain range:

Step 3: Enumerate Arithmetic Progression Size

Using standard progression length mapping tools:

Pattern Recognition

Finding pure integer steps matches finding values that fit LCM tracking parameters for base radical roots across whole block lengths.

Chapter Mix

Class 11 Mathematics: Binomial Theorem

Related Formula

Core Logic

Break the base parameter into a multiple of 10 form (1920 - 1$1920 - 1$) and use binomial expansions to decouple high factor blocks from final terminal digits.

Step 1: Set Up Binomial Expansion

Step 2: Isolate Low Factor Coefficients

All higher structural tracks sitting above index points present factors multiple loops over 100. Isolate trailing values:

Step 3: Deduce Final Trailing Quotient Product

Trailing structural indicators: 79$79$. Product calculation response: 7 × 9 = 63$7 \times 9 = 63$

Pattern Recognition

Using expansions around tens bases shifts focus entirely to the last two expansion expressions to find digit answers rapidly.

Chapter Mix

Class 11 Mathematics: Binomial Theorem