Please find the next THREE numbers in the patterns and explain how you got them:
1) 5, 5/2, 5/4, 5/8, 5/16....
2) 1, -3, 9, -27, 81...
3) 1/2, 1/2, 3/8, 1/4, 5/32...
4) -9, 101, -999, 10001, -99999...
1. each new number is 1/2 times the previous one ..
2. How about (-3)^n , were n = 0,1,2,3,...
3. What if I rewrote you fractions as
1/2 , 2/4, 3/8 , 4/16, 5/32 ...
Can you see what is happening ?
4. the odd-position numbers are all negative and contain as many 9's as the position in the sequence
e.g. the 7h number would be 9999999 (7 nines)
the even-position numbers are all positive and start and end with 1, with one less 0 between them as then position of the number
e.g. the 6th number is 1000001 (five 0's between the 1's)
5668
Sure! Let's find the next three numbers in each pattern:
Pattern 1: 5, 5/2, 5/4, 5/8, 5/16...
To get the next number, we divide the previous number by 2. Therefore, the pattern continues as follows:
Next Number 1: (5/16) ÷ 2 = 5/32
Next Number 2: (5/32) ÷ 2 = 5/64
Next Number 3: (5/64) ÷ 2 = 5/128
So, the next three numbers in the pattern are 5/32, 5/64, and 5/128.
Pattern 2: 1, -3, 9, -27, 81...
In this pattern, each number is multiplied by -3 to get the next number. Therefore, the pattern continues as follows:
Next Number 1: 81 × (-3) = -243
Next Number 2: -243 × (-3) = 729
Next Number 3: 729 × (-3) = -2187
So, the next three numbers in the pattern are -243, 729, and -2187.
Pattern 3: 1/2, 1/2, 3/8, 1/4, 5/32...
In this pattern, the numerator is an increasing sequence starting with 1, and the denominator is a sequence of powers of 2 starting with 2. Therefore, the pattern continues as follows:
Next Number 1: 5/32 × 3/2 = 15/64
Next Number 2: 15/64 × 3/2 = 45/128
Next Number 3: 45/128 × 3/2 = 135/256
So, the next three numbers in the pattern are 15/64, 45/128, and 135/256.
Pattern 4: -9, 101, -999, 10001, -99999...
In this pattern, each number alternates between a negative number and a positive number. The alternate numbers are obtained by multiplying the previous number by -1, and then adding the next odd integer cubed. Therefore, the pattern continues as follows:
Next Number 1: -99999 × (-1) + (11)^3 = 100010
Next Number 2: 100010 × (-1) + (13)^3 = -101021
Next Number 3: -101021 × (-1) + (15)^3 = 103940
So, the next three numbers in the pattern are 100010, -101021, and 103940.
I hope this helps!
1) To find the next three numbers in the pattern 5, 5/2, 5/4, 5/8, 5/16..., we can observe that each number is half of the previous number.
Starting with 5, we divide it by 2 to get 5/2. Then, we divide 5/2 by 2 to get 5/4. Continuing this process, we divide each subsequent number by 2 to get the next number in the pattern.
So, following this pattern, the next three numbers would be: 5/32, 5/64, 5/128.
2) The pattern in the sequence 1, -3, 9, -27, 81... can be realized by raising -3 to consecutive powers of 3.
Starting with 1, the next number is obtained by multiplying the previous number by -3.
1 * (-3) = -3
-3 * (-3) = 9
9 * (-3) = -27
Continuing the pattern, we can easily determine the next three numbers: -27 * (-3) = 81, 81 * (-3) = -243, -243 * (-3) = 729.
Therefore, the next three numbers in the pattern are: -243, 729, -2187.
3) The pattern in the sequence 1/2, 1/2, 3/8, 1/4, 5/32... can be identified by examining the numerator and the denominator.
The numerator remains the same in the first two terms, becomes double in the third term, double again in the fourth term, and continues this pattern for subsequent terms.
The denominator increases by a power of 2 in each term.
Based on these observations, we can find the next three numbers in the pattern:
The numerator for the fifth term will be double the numerator of the fourth term: 5/32 * 2 = 10/32.
The denominator for the fifth term will be double the denominator of the fourth term: 5/32 * 2 = 5/64.
Applying the same reasoning, we can find the numerator and denominator of the sixth and seventh terms.
The sixth term: 10/32 * 2 = 20/32, 5/64 * 2 = 5/128.
The seventh term: 20/32 * 2 = 40/32, 5/128 * 2 = 5/256.
Therefore, the next three numbers in the pattern are: 10/32, 5/64, 20/32, 5/128, 40/32, 5/256.
4) The pattern in the sequence -9, 101, -999, 10001, -99999... can be determined through an alternating series of negative and positive numbers.
Starting with -9, each subsequent number can be obtained by multiplying the previous number by -100 and then adding the corresponding positive number, which is obtained by raising 10 to the power equal to the position of the number in the sequence.
-9 * (-100) + 10^1 = 101
101 * (-100) + 10^2 = -999
-999 * (-100) + 10^3 = 10001
Continuing this pattern, we can find the next three numbers:
10001 * (-100) + 10^4 = -99999
-99999 * (-100) + 10^5 = 1000001
1000001 * (-100) + 10^6 = -9999999
Therefore, the next three numbers in the pattern are: -99999, 1000001, -9999999.