So there's a sequence:
0, 1/2, 3/4, 7/8, 15/16,...
and I have to find the pattern and the next two numbers. I don't understand how I should go about doing this. I don't believe it uses multiplication or division as it starts with 0. Help?
It looks like denominator is doubled each time, and the numerator is always one less than the denominator. From this, you should be able to find the next two numbers.
Notice the denominator is progressing at powers of 2:
2^1 = 2, 2^2 = 4, 2^3 = 8, 2^4 = 16.
Then the numerator is the denominator minus 1; therefore,
2^1 - 1 = 1, 2^2 - 1 = 3, 2^3 - 1 = 7, 2^4 - 1 = 15.
31/32
Ah, sequences, the never-ending source of mathematical puzzlement. Let's see if we can find the pattern in this one!
If we take a closer look at the sequence, we can observe that each term is obtained by adding 1 to the numerator and doubling the denominator compared to the previous term.
0, 1/2, 3/4, 7/8, 15/16,...
So, for example, to get from 0 to 1/2, we add 1 to the numerator (0 + 1 = 1) and double the denominator (2). And to get from 1/2 to 3/4, we add 1 to the numerator (1 + 1 = 2) and double the denominator (2 x 2 = 4).
Using this pattern, we can continue the sequence:
Next term: Add 1 to the numerator (15 + 1 = 16) and double the denominator (16 x 2 = 32).
Next, next term: Add 1 to the numerator (16 + 1 = 17) and double the denominator (32 x 2 = 64).
So, the next two numbers in the sequence are 16/32 and 17/64.
Remember, though, in the world of sequences, there's always a chance for a surprise twist! Keep your eyes peeled for any unexpected curveballs.
To find the pattern in the given sequence, it's helpful to look for any similarities or relationships between consecutive terms. Let's examine the differences between consecutive terms:
1st term: 0
2nd term: 1/2
Difference: 1/2 - 0 = 1/2
2nd term: 1/2
3rd term: 3/4
Difference: 3/4 - 1/2 = 1/4
3rd term: 3/4
4th term: 7/8
Difference: 7/8 - 3/4 = 1/8
From observing the differences, notice that the denominators of the differences follow a pattern: 2, 4, 8. It seems like the differences are powers of 2.
Now, let's examine the numerators of the differences:
1st difference: 1/2
2nd difference: 1/4
3rd difference: 1/8
The numerators form a decreasing sequence of powers of 2. This suggests that the numerators might be generated by starting with 1 and halving it repeatedly.
Now, let's try to form a general rule for generating the terms in the sequence. We can add each numerator to the previous term's denominator:
1st term: 0*2 + 1 = 1
2nd term: (1/2)*2 + 2 = 3/4
3rd term: (3/4)*2 + 4 = 7/8
4th term: (7/8)*2 + 8 = 15/16
Based on this pattern, we can generate the next two terms:
5th term: (15/16)*2 + 16 = 31/32
6th term: (31/32)*2 + 32 = 63/64
Therefore, the next two terms in the sequence are 31/32 and 63/64.