The first term of a sequence is `(1)/(2)` and the second term is `(1)/(4)`. Each term thereafter is the sum of the all the terms before it , If the `n^(th)` term is the first term of the sequence that is an integer , what is the value of n ?

Answer

To solve the problem, we need to find the value of \( n \) such that the \( n^{th} \) term of the sequence is the first integer. Let's break down the steps:

### Step 1: Define the sequence
The first term \( a_1 \) is given as \( \frac{1}{2} \) and the second term \( a_2 \) is \( \frac{1}{4} \).

### Step 2: Calculate subsequent terms
Each term thereafter is the sum of all the previous terms. We can express this mathematically as:
\[ a_n = \sum_{i=1}^{n-1} a_i \]

### Step 3: Calculate the third term \( a_3 \)
Using the definition:
\[ a_3 = a_1 + a_2 = \frac{1}{2} + \frac{1}{4} \]
To add these, we convert \( \frac{1}{2} \) to a fraction with a denominator of 4:
\[ a_1 = \frac{2}{4} \]
Now we can add:
\[ a_3 = \frac{2}{4} + \frac{1}{4} = \frac{3}{4} \]

### Step 4: Calculate the fourth term \( a_4 \)
Now we find \( a_4 \):
\[ a_4 = a_1 + a_2 + a_3 = \frac{1}{2} + \frac{1}{4} + \frac{3}{4} \]
Again, convert \( \frac{1}{2} \) to \( \frac{2}{4} \):
\[ a_4 = \frac{2}{4} + \frac{1}{4} + \frac{3}{4} = \frac{6}{4} = \frac{3}{2} \]

### Step 5: Calculate the fifth term \( a_5 \)
Next, we calculate \( a_5 \):
\[ a_5 = a_1 + a_2 + a_3 + a_4 = \frac{1}{2} + \frac{1}{4} + \frac{3}{4} + \frac{3}{2} \]
Convert \( \frac{3}{2} \) to a fraction with a denominator of 4:
\[ \frac{3}{2} = \frac{6}{4} \]
Now add:
\[ a_5 = \frac{2}{4} + \frac{1}{4} + \frac{3}{4} + \frac{6}{4} = \frac{12}{4} = 3 \]

### Step 6: Conclusion
The first integer in the sequence is \( a_5 \). Therefore, the value of \( n \) is \( 5 \).

### Final Answer
The value of \( n \) is \( 5 \).

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