To determine how much money Steven's grandmother has given him by the time he turns nine years old, we need to calculate the total amount of money given each year and then sum these amounts.
Let's break it down step by step:
1. First Birthday:
- Amount given: [tex]$1
2. Second Birthday:
- Amount given: $[/tex]2 (double of the first birthday)
3. Third Birthday:
- Amount given: [tex]$4 (double of the second birthday)
Each year, the amount is doubled from the previous year. We can represent the amount given on each birthday as:
- First Birthday: \( 2^0 = 1 \)
- Second Birthday: \( 2^1 = 2 \)
- Third Birthday: \( 2^2 = 4 \)
- ...
- Ninth Birthday: \( 2^8 = 256 \)
Now we need to sum these amounts to find the total amount of money given by the time Steven turned nine:
\[ 2^0 + 2^1 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8 \]
This is a geometric series where the first term \( a = 1 \) and the common ratio \( r = 2 \). For a geometric series, the sum of the first \( n \) terms can be calculated using the formula:
\[ S_n = \frac{a(r^n - 1)}{r - 1} \]
Here, \( n = 9 \), \( a = 1 \), and \( r = 2 \):
\[ S_9 = \frac{1(2^9 - 1)}{2 - 1} \]
\[ S_9 = 2^9 - 1 \]
\[ S_9 = 512 - 1 \]
\[ S_9 = 511 \]
Hence, by the time Steven turned nine years old, his grandmother had given him a total of $[/tex]511.
