If a sequence is as given below :
`1,1,2,3,5,8,13,21,34,….`
The unit digit of the 75 th term of this sequence will be:

Options

A. 0
B. 5
C. 7
D. none of these

Answer

To find the unit digit of the 75th term of the Fibonacci sequence, we can follow these steps:

### Step 1: Understand the Fibonacci Sequence
The Fibonacci sequence is defined as follows:
- The first two terms are 1 and 1.
- Each subsequent term is the sum of the two preceding terms.

So the sequence starts as:
1, 1, 2, 3, 5, 8, 13, 21, 34, ...

### Step 2: Identify the Fibonacci Formula
The nth term of the Fibonacci sequence can be calculated using the formula:
\[ F(n) = F(n-1) + F(n-2) \]
where \( F(1) = 1 \) and \( F(2) = 1 \).

### Step 3: Calculate the 75th Term
To find the 75th term, we can use the formula iteratively:
- \( F(3) = F(2) + F(1) = 1 + 1 = 2 \)
- \( F(4) = F(3) + F(2) = 2 + 1 = 3 \)
- \( F(5) = F(4) + F(3) = 3 + 2 = 5 \)
- Continuing this process up to \( F(75) \).

However, calculating all terms up to 75 is not efficient for just finding the unit digit.

### Step 4: Focus on the Unit Digits
Instead of calculating the entire Fibonacci numbers, we can focus only on the unit digits of each term. The unit digits of the Fibonacci sequence repeat every 60 terms. This means we can find the unit digit of the 75th term by finding the position of 75 modulo 60.

### Step 5: Calculate 75 Modulo 60
\[ 75 \mod 60 = 15 \]
This means the unit digit of the 75th term is the same as the unit digit of the 15th term.

### Step 6: Find the Unit Digit of the 15th Term
Now, we need to find the unit digit of the 15th term:
1. Calculate the Fibonacci sequence up to the 15th term:
- \( F(1) = 1 \)
- \( F(2) = 1 \)
- \( F(3) = 2 \)
- \( F(4) = 3 \)
- \( F(5) = 5 \)
- \( F(6) = 8 \)
- \( F(7) = 13 \) (unit digit = 3)
- \( F(8) = 21 \) (unit digit = 1)
- \( F(9) = 34 \) (unit digit = 4)
- \( F(10) = 55 \) (unit digit = 5)
- \( F(11) = 89 \) (unit digit = 9)
- \( F(12) = 144 \) (unit digit = 4)
- \( F(13) = 233 \) (unit digit = 3)
- \( F(14) = 377 \) (unit digit = 7)
- \( F(15) = 610 \) (unit digit = 0)

### Step 7: Conclusion
The unit digit of the 75th term of the Fibonacci sequence is the same as the unit digit of the 15th term, which is 0.

### Final Answer
The unit digit of the 75th term of the sequence is **0**.

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