The sequence can be written as
As you can see, every term is 5 times a power of 2. To be precise, the nth term is 5 times the (3 - n)th power of 2, or
Okay this is a series and sequence equation.
I do not know how to prove that the following equation can be wrote in that way. The question is:
20;10;5; .... is a G.S.
Show that the n^th term of the sequence is given by the formula:
Please help me.
In answer to how I realized the sequence could be written as
, , , ...
it's a matter of pattern recognition. I noticed any given term was one-half the previous term, so I knew powers of 2 were involved ( ). I also noticed the first three terms were multiples of 5, so I knew 5 was involved. Putting it all together, I re-wrote each term as 5 times a power of 2.
In answer to why the exponent is , consider the exponents of the first three terms. For the 1st term, the exponent is 2. For the 2nd term, the exponent is 1. For the 3rd term, the exponent is 0. In other words, the exponent decreases in successive terms. Thus, is appropriate because decreases as increases.
The in the exponent is necessary to "align" the terms and the exponents. If it weren't there, then the exponent of the 1st term would be -1. However, the correct exponent of the 1st term is 2, so ask yourself what could you add to -1 to produce 2. The answer is 3. Thus, the correct exponent is .