What is the 16th in the Progression 1 4 16

When dealing with mathematical sequences, one often encounters a specific type known as a geometric progression. A geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In the sequence '1 4 16', we can see this pattern clearly:

Identifying the Common Ratio

Let's break down the progression and identify the common ratio:

The first term is 1. The second term is 4. The ratio between the second and first term is 4/1 4. The third term is 16. The ratio between the third and second term is 16/4 4.

As we can see, the common ratio (r) is 4. Now, let's delve into finding the 16th term in this sequence.

Calculating the 16th Term of the Sequence

The general formula for the nth term of a geometric progression is:

[a_n ar^{n-1}]

Here, (a) is the first term of the sequence and (r) is the common ratio. For the 16th term, we have:

[a_{16} 1 times 4^{15}]

Let's calculate (4^{15}) to find the 16th term:

Step-by-Step Calculation

[4^1 4] [4^2 4 times 4 16] [4^3 4 times 16 64] [4^4 4 times 64 256] [4^5 4 times 256 1024] [4^6 4 times 1024 4096] [4^7 4 times 4096 16384] [4^8 4 times 16384 65536] [4^9 4 times 65536 262144] [4^{10} 4 times 262144 1048576] [4^{11} 4 times 1048576 4194304] [4^{12} 4 times 4194304 16777216] [4^{13} 4 times 16777216 67108864] [4^{14} 4 times 67108864 268435456] [4^{15} 4 times 268435456 1073741824]

The 16th Term of the Sequence

Therefore, the 16th term of the sequence, which is (a_{16}), is:

[a_{16} 1 times 1073741824 1073741824]

Expressed in terms of powers of 2, since (4 2^2), we can write:

[4^{15} (2^2)^{15} 2^{30}]

So, the 16th term is (2^{30}).

Repetition and Pattern in Multiplying by 4

Let's further illustrate the multiplication pattern for clarity:

[1 times 4 4] [4 times 4 16] [4 times 16 64] [16 times 64 1024] [64 times 1024 65536]

Continuing this process further would follow the same pattern of multiplying each term by 4 to get the next term in the sequence.

Conclusion

Through the use of the geometric progression formula, we calculated the 16th term in the sequence to be (2^{30}) or 1073741824. This demonstrates the power of recognizing patterns and applying mathematical formulas to solve complex problems efficiently.

Understanding geometric progressions can be incredibly useful in various fields including algebra, computer science, and even finance. By mastering these concepts, one can solve a wide range of real-world problems.