Question

How many dots will figure n have? Explain how you know this.

Answer 1

Up to the table everything is right at the exception of the n value (in the table).

How many dots... figure 10: You found it it's 49 (and not 9)

How many dots... figure 25: You found it it's 109 (and not 24). Expanding it gives : 13+24x4

How many dots... figure 100: You found it it's 409 (and not 99). Expanding it gives : 13+99x4

How many common difference are added to 13 for figure 100. It's easy you also found it, it's 99, the common difference being 4 so it's 99x4

Now let's calculate the number of dots in figure "n" which is:

a(n) = a₁ + (n-1)x4 (4 being the common difference and a₁ = 1st term)

explanation: lookup again at your table and compare the FIG # part with the EXPAND part:

FIG # EXPAND (and notice the figures in the parenthesis)
--------- -----------
1 13 + (0)x4 = 13 (for fig# 1 we have (0)
2 13 + (1)x4 = 17 (for fig# 2 we have (1)
3 13 + (2)x4 = 21 (for fig# 3 we have (2)
4 13 + (3)x4 = 25 (for fig# 4 we have (3)
. ........................
. .........................
. ........................
n 13 + (n-1)x4 (for fig# we have (n-1)

As you notice the number in the parenthesis is always = fig # - 1

Hope that you understand this formula of an arithmetic progression with first term a = 13 and 4, the common difference. Note that n is the number of terms

Answer 2

n = ((n-1) + 4) + 4, to my understanding.
although your answers seem right written in the columns, but your method of applying numbers to equation seems incorrect in 'expand' column. for example, answer for 3 should be something like this...
3 21 17+4
4 25 21+4
hope it helps.