To solve the problem of finding the value of \( n \) such that the \( 100^{th} \) term of the series \( 3 + 8 + 22 + 72 + 266 + 1036 + \ldots \) is divisible by \( 2^n \), we will follow these steps:
### Step 1: Identify the Pattern in the Series
We start by calculating the first differences of the series:
- \( 8 - 3 = 5 \)
- \( 22 - 8 = 14 \)
- \( 72 - 22 = 50 \)
- \( 266 - 72 = 194 \)
- \( 1036 - 266 = 770 \)
So, the first differences are \( 5, 14, 50, 194, 770 \).
### Step 2: Calculate the Second Differences
Next, we calculate the second differences:
- \( 14 - 5 = 9 \)
- \( 50 - 14 = 36 \)
- \( 194 - 50 = 144 \)
- \( 770 - 194 = 576 \)
The second differences are \( 9, 36, 144, 576 \).
### Step 3: Identify the Pattern in the Second Differences
The second differences appear to form a geometric series. We can see that:
- \( 36 = 4 \times 9 \)
- \( 144 = 4 \times 36 \)
- \( 576 = 4 \times 144 \)
This suggests that the second differences are multiplied by \( 4 \).
### Step 4: Formulate the General Term
We can express the \( n^{th} \) term of the series in a general form. The \( n^{th} \) term can be represented as:
\[
T_n = a \cdot 4^{n-1} + b \cdot n + c
\]
### Step 5: Set Up the System of Equations
Using the first few terms of the series, we can set up equations to solve for \( a \), \( b \), and \( c \):
1. For \( n = 1 \): \( a + b + c = 3 \)
2. For \( n = 2 \): \( 4a + 2b + c = 8 \)
3. For \( n = 3 \): \( 16a + 3b + c = 22 \)
### Step 6: Solve the System of Equations
From the first equation, we can express \( c \) in terms of \( a \) and \( b \):
\[
c = 3 - a - b
\]
Substituting \( c \) into the second equation:
\[
4a + 2b + (3 - a - b) = 8 \implies 3a + b = 5 \quad \text{(Equation 1)}
\]
Substituting \( c \) into the third equation:
\[
16a + 3b + (3 - a - b) = 22 \implies 15a + 2b = 19 \quad \text{(Equation 2)}
\]
Now we solve Equations 1 and 2 simultaneously.
From Equation 1, we can express \( b \):
\[
b = 5 - 3a
\]
Substituting into Equation 2:
\[
15a + 2(5 - 3a) = 19 \implies 15a + 10 - 6a = 19 \implies 9a = 9 \implies a = 1
\]
Now substituting \( a = 1 \) back into Equation 1:
\[
3(1) + b = 5 \implies b = 2
\]
Finally, substituting \( a \) and \( b \) back to find \( c \):
\[
c = 3 - 1 - 2 = 0
\]
### Step 7: Write the General Term
Thus, the \( n^{th} \) term of the series is:
\[
T_n = 4^{n-1} + 2n
\]
### Step 8: Find the \( 100^{th} \) Term
Now, we find \( T_{100} \):
\[
T_{100} = 4^{99} + 2 \cdot 100 = 4^{99} + 200
\]
### Step 9: Determine Divisibility by \( 2^n \)
We need to find the highest power of \( 2 \) that divides \( T_{100} \).
1. \( 4^{99} = (2^2)^{99} = 2^{198} \)
2. \( 200 = 2^3 \cdot 25 \)
Now, we can express \( T_{100} \):
\[
T_{100} = 2^{198} + 2^3 \cdot 25
\]
Factoring out \( 2^3 \):
\[
T_{100} = 2^3(2^{195} + 25)
\]
### Step 10: Check the Divisibility
Now, we need to check if \( 2^{195} + 25 \) is even or odd. Since \( 2^{195} \) is even and \( 25 \) is odd, their sum is odd. Therefore, \( 2^{195} + 25 \) is not divisible by \( 2 \).
Thus, the highest power of \( 2 \) that divides \( T_{100} \) is \( 3 \).
### Final Answer
The value of \( n \) such that \( T_{100} \) is divisible by \( 2^n \) is:
\[
\boxed{3}
\]
