Rom

What are these questions?

Are you supposed to be solving them by hand within a reasonable amount of time?

It's pretty trivial solving them with software.

Thoughts?

\(\text{I guess the series they are after has $a=64,~r=\dfrac 1 2$}\\ 64 \sum \limits_{k=0}^8\left(\dfrac 1 2 \right)^k = 64 \dfrac{1-\left(\frac 1 2 \right)^9}{1 - \frac 1 2} = \\ 64 \dfrac{1 - \frac{1}{512}}{\frac 1 2} = \dfrac{511}{4}=127.75\\ \text{Well done!}\)

\(\text{well then you should know that the sum of the first three terms are}\\ a + \dfrac 2 3 a + \dfrac 4 9 a = \dfrac{19}{9}a = 171\\ a=81\)

Each column in the first row has 10 possible entries.

There are 5 columns

10⁵ = 100000

first off do you know what a geometric series is?

the line that marks the top of the triangle for x<0 is given by 

y = 5+x/2

we can list the points such that y>0, y< 5+x/2

let's list the y boundary for each of the values -5 < x < 0

(-9, 0.5), (-8, 1), (-7, 1.5), (-6, 2), (-5, 2.5), (-4, 3), (-3, 3.5), (-2, 4), (-1, 4.5)

so we see the following points are in the interior for x<0

(-7, 1), (-6, 1), (-5, 1), (-5, 2), (-4,1), (-4, 2), (-3, 1), (-3, 2), (-3, 3), (-2, 1), (-2, 2), (-2, 3), (-1, 1), (-1, 2), (-1, 3), (-1, 4)

16 points.  There are 16 simllar points for x>0.  That brings us to 32 points

Then on the y axis there is (0,1), (0,2), (0,3), (0,4) which brings the total number to 36.

I'd write all the numbers less than 20 that can be written as the sum of two primes.

Then sum all the rest.  Note that one prime is not the sum of two primes.

2,3,5,7,11,13,17 are the primes of interest

5, 7, 9, 13, 15, 19 are sums including 2

8, 10, 14, 16, 20 include 3

12, 16, 18 include 5

18, 20

and that's it.  That leaves

1, 2, 3, 4, 6, 11, 17

These sum to 34

You were on the right track.  You just didn't see it through.

All the numbers are valid first numbers so write them down on a page with some space between them.

Then start listing valid numbers for each below them.

For example 3, 4, 5 are all valid following a 1

Just continue that for 4 rows.  You'll also use the fact that codes must have distinct numbers to trim branches as you go.

At the bottom you'll have the leaves from the survivors.  Count them.  There are 20.

just make a tree

maybe look it up in the dictionary?