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

1 Answer
May 9, 2016

Answer:

#(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 #7#th row (calling the first row the #0#th). Personally I prefer to call it the #8#th 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#