1,1,2,4,16,128,4096
What are the next 3 terms.
This one was quite easy. This is what to do. Write the numbers down and start looking at any relationships between consecutive numbers.
In this case look at the ratios between each consecutive numbers. For the next term consider the relationship between the previous two ratios.
$\displaystyle 1 {\color{red}\times 1 = } , 1 {\color{red}\times 2 = } , 2 {\color{red}\times 2 = } , 4 {\color{red}\times 4 = } , 16 {\color{red}\times 8 = } , 128 {\color{red} \times 32 = } ,\dots $
Ohrly? Interesting.
I was playing around to find a number pattern so I started with pascals triangle.
$\displaystyle
\binom{n}{k}$
then I added up each row, but not all rows. I chose a row correlating to a financing number subtracted by 1.
so
$\displaystyle
\binom{(F_{n}+F_{n-1})-1}{k}$
which gives
$\displaystyle \binom{1-1}{k}$
$\displaystyle \binom{1-1}{k}$
$\displaystyle \binom{2-1}{k}$
$\displaystyle \binom{3-1}{k}$
ect.
But I wanted the sum so.
$\displaystyle \sum^{F_n-1}_{k}\binom{F_n-1}{k}$
$\displaystyle \left [ \sum^{0}_{k}\binom{0}{k} \right ]=1$
$\displaystyle \left [ \sum^{0}_{k}\binom{0}{k} \right ]=1$
$\displaystyle \left [\sum^{1}_{k}\binom{1}{k}\right ]=2$
$\displaystyle \left [ \sum^{2}_{k}\binom{2}{k}\right ]=4$
$\displaystyle \left [ \sum^{4}_{k}\binom{4}{k}\right ]=16$
$\displaystyle \left [ \sum^{7}_{k}\binom{7}{k}\right ]=128$
While your method works yes (although I did not think of it)
It does not explain the first term, only all after.
1,1,2,4,16
So, you're answers work but are technically incorrect.
$\displaystyle
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