How do you use the binomial theorem to approximate the value of 1.07^7 ?

1 Answer
George C.
May 9, 2016

(1.07)^7 ~~ 1.60578

Explanation:

By the binomial theorem we have:

(a+b)^7 = sum_(k=0)^7 ((7),(k)) a^(7-k)b^k

where ((7),(k)) = (7!)/(k!(7-k)!)

We can get these binomial coefficients from the row of Pascal's triangle that begins 1, 7. Some people call this the 7th row (calling the first row the 0th). Personally I prefer to call it the 8th row, but regardless of what you call it, it's the one that begins 1, 7:

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So:

(a+b)^7 = a^7+7a^6b+21a^5b^2+35a^4b^3+35a^3b^4+21a^2b^5+7ab^6+b^7

Putting a=1 and b=0.07 we have:

1.07^7 ~~ 1+7(0.07)+21(0.07)^2+35(0.07)^3+35(0.07)^4+21(0.07)^5

=1+7(0.07)+21(0.0049)+35(0.000343)+35(0.00002401)+21(0.0000016807)

=1+0.49+0.1029+0.012005+0.00084035+0.0000352947

~~1.60578

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