Use inductive reasoning to predict the next number in a. 1,4,9,16,25,36,49,?

Result:
(a)$64$ (b)$53$ (c)$\frac{7}{8}$ (d)$47$
Solution:
a. The given sequence is: $1,4,9,16,25,36,49,?$
Observing the pattern, we can see that each number in the sequence is the square of its position in the sequence. Hence, the next number would be the square of the next position, which is $8$.
Therefore, the next number in the sequence is $64$.
b. The given sequence is: $1,5,12,22,35,?$
To find the pattern, we can observe that each number in the sequence is obtained by adding the position of the number in the sequence multiplied by $3$, starting from $1$.
Using this pattern, the next number in the sequence would be $35+(6\times 3)=53$.
Therefore, the next number in the sequence is $53$.
c. The given sequence is: $\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5},\frac{5}{6},\frac{6}{7},?$
Looking at the pattern, we can see that each number in the sequence is obtained by incrementing the numerator and denominator by $1$.
Using this pattern, the next number in the sequence would be $\frac{6}{7}+\frac{1}{1}=\frac{7}{8}$.
Therefore, the next number in the sequence is $\frac{7}{8}$.
d. The given sequence is: $3,5,9,15,23,33,?$
Observing the pattern, we can see that each number in the sequence is obtained by adding the position of the number in the sequence multiplied by $2$, starting from $1$.
Using this pattern, the next number in the sequence would be $33+(7\times 2)=47$.
Therefore, the next number in the sequence is $47$.

Step 1:
a. The given sequence is 1, 4, 9, 16, 25, 36, 49. We can observe that each term is a perfect square: $1=1^2$, $4=2^2$, $9=3^2$, $16=4^2$, $25=5^2$, $36=6^2$, $49=7^2$.
Using inductive reasoning, we can conclude that the next number will be the square of the next integer. So, the next number in the sequence is $8^2=64$. Therefore, the complete sequence is $1,4,9,16,25,36,49,\mathbf{64}$.
Step 2:
b. The given sequence is 1, 5, 12, 22, 35. By examining the differences between consecutive terms, we can see that the differences are increasing by 1 each time: $5-1=4$, $12-5=7$, $22-12=10$, $35-22=13$.
To find the next difference, we add 1 to the previous difference: $13+1=14$. Now, we can find the next term by adding the next difference to the last term: $35+14=49$.
Therefore, the next number in the sequence is $\mathbf{49}$. The complete sequence is $1,5,12,22,35,\mathbf{49}$.
Step 3:
c. The given sequence is $\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5},\frac{5}{6},\frac{6}{7}$. By observing the pattern, we can see that the numerator is increasing by 1, while the denominator is increasing by 1 as well.
To find the next number, we can continue this pattern: $\frac{6}{7}+\frac{1}{1}=\frac{7}{8}$.
Therefore, the next number in the sequence is $\frac{\mathbf{7}}{\mathbf{8}}$. The complete sequence is $\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5},\frac{5}{6},\frac{6}{7},\frac{\mathbf{7}}{\mathbf{8}}$.
Step 4:
d. The given sequence is 3, 5, 9, 15, 23, 33. By observing the differences between consecutive terms, we can notice that the differences are increasing by 2 each time: $5-3=2$, $9-5=4$, $15-9=6$, $23-15=8$, $33-23=10$.
To find the next difference, we add 2 to the previous difference: $10+2=12$. Now, we can find the next term by adding the next difference to the last term: $33+12=45$.
Thus, the next number in the sequence is $\mathbf{45}$. The complete sequence is $3,5,9,15,23,33,\mathbf{45}$.

a. The given sequence is 1, 4, 9, 16, 25, 36, 49. By observing the sequence, we can see that each number is the square of its corresponding position in the sequence. In other words, the nth term is given by $n^2$. Therefore, the next number in the sequence is $8^2=64$.
b. The given sequence is 1, 5, 12, 22, 35. By observing the differences between consecutive terms, we can see that they form a pattern: 4, 7, 10, 13. This indicates that the second difference is constant, which means the sequence is likely quadratic. Let's denote the position in the sequence as n. Then, the nth term can be expressed as $a{n}^2+bn+c$.
To find the coefficients a, b, and c, we can substitute the values of n and the corresponding terms in the sequence into the equation. By solving the resulting system of equations, we can determine the values of a, b, and c. However, in this case, I will use the formula for the nth term of a quadratic sequence, which is $a{n}^2+(a-1)n+1$. By substituting n = 6 into the formula, we can find the next term as follows:
$6^2+(6-1)6+1=36+5\times 6+1=36+30+1=67$
Therefore, the next number in the sequence is 67.
c. The given sequence is $\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5},\frac{5}{6},\frac{6}{7}$. By observing the pattern, we can see that the numerator increases by 1 while the denominator increases by 1, starting from $\frac{1}{2}$. Therefore, the next number in the sequence is $\frac{7}{8}$.
d. The given sequence is 3, 5, 9, 15, 23, 33. By observing the differences between consecutive terms, we can see that they form a pattern: 2, 4, 6, 8, 10. This indicates that the differences between consecutive terms are increasing by 2 each time. Therefore, we can predict that the next difference will be 12.
To find the next number, we add the next difference to the last term in the sequence: $33+12=45$. Therefore, the next number in the sequence is 45.