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AP EAMCET2010MathematicsBinomial Theorem

The numbers a_n=6^n-5 n for n=1,2,3, when divided by 25 leave the remainder

Options

  1. A9
  2. B7
  3. C3
  4. D1

Correct answer

D. 1

Step-by-step solution

Given, a_n=6^n-5 n, n=1,2,3, We take; 6^n=(1+5)^n Expand with binomial expansion 6^n= ^n C₀+ ^n C₁ 5+ ^n C₂ 5^2+ ^n C₃ 5^3+ 6^n=1+n 5+ ^n C₂ 25+ ^n C₃ 5^3+ (6^n-5 n )=1+25 ^n C₂+ ^n C₃ 5+ (6^n-5 n )=1+25 k where k= positive integer. Hence, a_n=6^n-5 n divided by 25 and leave the remainder =1

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