Grade 12Algebra
What is the binomial expansion of a^n -b^n. . ...... . .
1 Answer
Samyak Jain
an - bn = an [1 – (b/a)]
(b/a) is the common ratio r of the finite geometric series :
1 + r + r2 + r3 + ….............. + rn-1 , where r = (b/a)
1 + r + r2 + r3 + ….............. + rn-1 = (1 – rn)/ (1 – r)
(1 – rn) = (1 + r + r2 + r3 + ….............. + rn-1) (1 – r)
i.e. 1 – (b/a) = {1 + (b/a) +(b/a)2 + (b/a)3 + ….............. + (b/a)n-1} {1 – (b/a)}
1 – (b/a) = {1 + (b/a) +(b/a)2 + (b/a)3 + ….............. + (b/a)n-1} {a – b} / a
Therefore, an - bn = an {1 + (b/a) +(b/a)2 + (b/a)3 + ….............. + (b/a)n-1} {a – b} / a
an - bn = an-1 {1 + (b/a) +(b/a)2 + (b/a)3 + ….............. + (b/a)n-1} {a – b}
= (a – b) (an-1 + an-2 b + an-3 b2 + a bn-2 + bn-1)
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