Consider the sequence `1,3,3,3,5,5,5,5,5,7,7,7,7,7,7,7,……` and evaluate its `2016 ^(th)` term.

Answer

To find the 2016th term of the sequence `1, 3, 3, 3, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 7, ...`, we can follow these steps:

### Step 1: Identify the pattern in the sequence
The sequence consists of odd numbers where:
- The number `1` appears `1` time.
- The number `3` appears `3` times.
- The number `5` appears `5` times.
- The number `7` appears `7` times.
- And so on...

### Step 2: Determine how many terms are generated by the odd numbers
For any odd number `n`, it appears `n` times. The total number of terms contributed by the first `k` odd numbers is given by the formula:
\[ S_k = 1 + 3 + 5 + ... + (2k - 1) = k^2 \]
This means that the `k`th odd number contributes `k^2` terms to the sequence.

### Step 3: Find the largest `k` such that `k^2` is less than or equal to `2016`
We need to find the largest integer `k` such that:
\[ k^2 \leq 2016 \]

Calculating:
- \( 44^2 = 1936 \)
- \( 45^2 = 2025 \)

Since \( 1936 \leq 2016 < 2025 \), we find that \( k = 44 \) is the largest integer satisfying the condition.

### Step 4: Determine the number of terms up to the 44th odd number
The total number of terms contributed by the first 44 odd numbers is:
\[ S_{44} = 44^2 = 1936 \]

### Step 5: Calculate the remaining terms to reach the 2016th term
Now, we need to find how many more terms are needed to reach the 2016th term:
\[ 2016 - 1936 = 80 \]

### Step 6: Identify the next odd number and its contribution
The next odd number after the first 44 odd numbers is `89` (which is the 45th odd number). This number appears `45` times.

### Step 7: Determine the 2016th term
Since we need 80 more terms and the number `89` will provide the next `45` terms, we can conclude:
- The 1937th term to the 1981st term will be `89`.
- We still need `80 - 45 = 35` more terms.

The next number after `89` is `91`, which will appear `47` times. Therefore, the 1982nd term to the 2028th term will be `91`.

Thus, the 2016th term is `91`.

### Final Answer
The 2016th term of the sequence is **91**.

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