Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
AP EAMCET201921 Apr 2019Morning ShiftMathematicsBinomial TheoremActual

Let a₀, a₁, a₂, a_n R be in an arithmetic progression and let C₀, C₁, C₂, , C_n be the binomial coefficients. Then _ k=0 ^n a_k C_k=

Options

  1. A1 2 (a₀+a_n )
  2. B(a₀+a_n ) 2^ n-1
  3. C(a₀+a_n )
  4. D0

Correct answer

B. (a₀+a_n ) 2^ n-1

Step-by-step solution

aligned _ k=0 ^n a_k C_k=a₀ C₀+a₁ C₁+a₂ C₂ & + +a_n C_n =a₀ C₀+ (a₀+d ) C₁ & + (a₀+2 d ) C₂ & + + (a₀+n d ) C_n aligned Where d is an common difference of an AP aligned = & a₀ (C₀+C₁+ +C_n )+ & d (C₁+2 C₂+3 C₃+ + ^n C_n ) = & a₀ 2^n+d (n 2^ n-1 ) = & 2^ n-1 [2 a₀+n d ]= (a₀+a_n ) 2^ n-1 aligned

Practice Binomial Theorem on Quantrex Academy →

More from Binomial Theorem

If n 13 , n 14 and n 15 are in arithmetic progression, then the positive integer value of ' n ' can be 2026If the coefficients of x^2 and x^3 in the expansion of (3 + kx)^9 are equal, then the value of ' k ' is 2026The remainder when 7¹⁰³ is divided by 25 is 2026_ r=1 ¹⁵ r^2 ( ¹⁵ C_r 15 r-1 )= 20251 81^ n - ^ 2 n C ₁ 10 81^ n + ^ 2 n C ₂ 10^2 81^ n - + 10^ 2 n 81^ n = 2025If x is positive real number and the first negative term in the expansion of (1+ x )^ 27 / 5 is t _ k then k = 2025In the binomial expansion of (p-q)¹⁴ , if the sum of 7^ th term and 8^ th term is zero, then p+q p-q = 2025The numerically greatest term in the expansion of (x+3 y)¹³ , when x= 1 2 and y= 1 3 is 2025 Full Binomial Theorem list All AP EAMCET PYQs