Unraveling the Pattern and Finding the 100th Term in the Sequence 124711

The sequence you have provided is: 1 2 4 7 11. At first glance, this sequence may seem like a chaotic array of numbers; however, upon closer inspection, it follows an interesting pattern.

Understanding the Pattern

To identify the pattern, let's examine the differences between consecutive terms:

2 - 1 1

4 - 2 2

7 - 4 3

11 - 7 4

The differences are: 1 2 3 4, which indicates that the difference between terms increases by 1 each time. This suggests a quadratic growth pattern.

Deriving the Formula for the nth Term

We can derive a formula for the nth term of the sequence using the differences we identified:

Step-by-Step Derivation

The first term ( a_1 1 ). The second term ( a_2 a_1 1 2 ). The third term ( a_3 a_2 2 4 ). The fourth term ( a_4 a_3 3 7 ). The fifth term ( a_5 a_4 4 11 ).

Continuing this pattern, we can express the nth term as:

$a_n a_{n-1} n-1$

This can be rewritten as:

$a_n a_1 sum_{k1}^{n-1} k$

The sum of the first m integers is given by the formula:

$sum_{k1}^{m} k frac{m(m-1)}{2}$

Substituting this into our formula:

$a_n 1 sum_{k1}^{n-1} k 1 frac{n-1(n-1)}{2}$

After simplifying, we get:

$a_n 1 frac{n-1(n-1)}{2} frac{n-1(n-1)}{2} 1$

Finding the 100th Term

To find the 100th term, we use the formula we derived:

$a_{100} 1 frac{100-1 cdot 100}{2}$

Calculating the expression inside the formula:

$a_{100} 1 frac{99 cdot 100}{2} 1 frac{9900}{2} 1 4950 4951$

Therefore, the 100th number in the sequence is 4951.

Alternative Approach and Pattern Recognition

There's an alternative way to visualize the pattern in the sequence:

1st term: 1
2nd term: 1 1
3rd term: 1 1 2
4th term: 1 1 2 3
5th term: 1 1 2 3 4
…
nth term: 1 1 2 3 4 … n-1

The nth term of the sequence is the number formed by the sum of the first n-1 natural numbers plus 1:

$a_n 1 frac{n-1(n-1)}{2}$

For the 100th term:

$1 frac{100-1 cdot 100}{2} 1 frac{99 cdot 100}{2} 1 4950 4951$

This confirms our earlier calculation.

Conclusion

By understanding the pattern and applying the appropriate mathematical principles, we can accurately find the 100th term of the given sequence:

$a_{100} 4951$

This exercise not only helps us recognize the underlying mathematical growth but also demonstrates the power of pattern recognition in solving complex problems.