Given $\log_{10}2 = m$ and $\log_{10}3 = n$, find $\log_{10}24$ in terms of $m$ and $n$.

Problem 4:

1. Problem Description

Given $\log_{10}2 = m$ and $\log_{10}3 = n$, find $\log_{10}24$ in terms of $m$ and $n$.

2. Solution Steps

We want to express $\log_{10}24$ in terms of $m$ and $n$.

First, we find the prime factorization of 24:

$24 = 2^3 \cdot 3$.

Now, we can use the properties of logarithms:

$\log_{10}24 = \log_{10}(2^3 \cdot 3)$

$\log_{10}24 = \log_{10}(2^3) + \log_{10}3$

$\log_{10}24 = 3\log_{10}2 + \log_{10}3$

Since $\log_{10}2 = m$ and $\log_{10}3 = n$, we have:

$\log_{10}24 = 3m + n$

3. Final Answer

$3m + n$

Problem 5:

1. Problem Description

Find the 5th term of the sequence: 2, 5, 10, 17, ...

2. Solution Steps

Let's analyze the differences between consecutive terms:

5 - 2 = 3

10 - 5 = 5

17 - 10 = 7

The differences are increasing by 2 each time. This suggests that the sequence is quadratic. The next difference should be 7 + 2 =

9. So the next term in the sequence will be 17 + 9 =

2

6. Therefore, the 5th term is

2

6.

Another approach: The general term of the sequence is $a_n = n^2 + 1$.

For n=1: $1^2 + 1 = 2$

For n=2: $2^2 + 1 = 5$

For n=3: $3^2 + 1 = 10$

For n=4: $4^2 + 1 = 17$

For n=5: $5^2 + 1 = 26$

3. Final Answer

26