How do you write a convincing argument to show why #3^0=1# using the following pattern: #3^5=243, 3^4=81, 3^3=27, 3^2=9#?

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Answer 1 · 2017-09-26T11:36:30

Refer to the Explanation.

#3^5,3^4,3^3,3^2,3^1,3^0,...#

#=243, 81=243/3, 27=81/3, 9=27/3, 3=9/3, 1=3/3.#

# :. 3^0=1.#

Answer 2 · 2017-09-26T11:47:53

See argument below

The most convincing argument I can think of off the bat is simply:

#x^0 = 1 forall x in RR#

However, I guess that was not the argument being sought here!

Here we have a sequence: #a_n=3^n# for #n= 5 to 2#

We can extend this to: #a_1 = 3^1 = 3#

Clearly: #a_(n-1)/a_n = 1/3 -> a_(n-1) = a_n/3#

Now consider #a_0 =a_1/3 = 3/3 =1#

Since #a_0 -= 3^0# the proposition is proved.