Question

Let Tn, be the number of dots in figure.

How many dots are there in figure 30?

Look at the picture (don't mind of the highlighted ones).​

Answer 1

4294967293 dots

Explanation:

Figure 1 has 5 dots.

Figure 2 has 13 dots.

Figure 3 has 29 dots.

Subtract and find different of each term:

• 13-5 = 8
• 29-13 = 16

Let
`\displaystyle \large{b_n}` be sequence of difference (and the sequence appears to be geometric.)

So for
`\displaystyle \large{b_n}`, find the common ratio which is 2.

Hence,
`\displaystyle \large{b_n = 8(2)^(n-1)}` - recall geometric sequence formula below:

`\displaystyle \large{a_n = a_1r^(n-1)}`

The sequence of difference
`\displaystyle \large{b_n=8(2)^(n-1)}` can be simplified to:

`\displaystyle \large{b_n=2^3(2)^(n-1)}\\\displaystyle \large{b_n=2^(n+2)}`

Now to find the original sequence:

`\displaystyle \large{a_n = a_1 + \sum_(k=1)^(n-1)b_k}`

Hence:

`\displaystyle \large{T_n=5+\sum_(k=1)^(n-1)8(2)^(k-1)}`

Recall:

`\displaystyle \large{\sum_(k=1)^(n-1) a_1r^(n-1) =a_1\left( (1-r^(n-1))/(1-r)\right)}`

Therefore:

`\displaystyle \large{T_n=5+\sum_(k=1)^(n-1)8(2)^(k-1)}\\\displaystyle \large{T_n=5+8\left((1-2^(n-1))/(1-2)\right)}\\\displaystyle \large{T_n=5+8\left((1-2^(n-1))/(-1)\right)}\\\\\displaystyle \large{T_n=5+8(-1+2^(n-1))}\\\displaystyle \large{T_n=5-8+8(2)^(n-1)}\\\displaystyle \large{T_n=2^(n+2)-3}`

Therefore, the sequence for 5,13,29 is
`\displaystyle \large{2^(n+2)-3}`.

Therefore, in figure 30:

`\displaystyle \large{T_(30)=2^(30+2)-3}\\\displaystyle \large{T_(30)=2^(32)-3}\\\displaystyle \large{T_(30)=4294967293}`

Therefore, there are 4294967293 dots in figure 30