To find the three consecutive odd integers such that the sum of the first, two times the second, and three times the third is 70, let's define the integers and set up the equation.

Let the three consecutive odd integers be:

• $x$ (the first odd integer)
• $x + 2$ (the second odd integer)
• $x + 4$ (the third odd integer)

According to the problem, the sum of the first, two times the second, and three times the third is 70. This gives us the equation: $x + 2(x + 2) + 3(x + 4) = 70$

Now, let's solve this equation step by step:

1. Expand the equation: $x + 2x + 4 + 3x + 12 = 70$

2. Combine like terms: $x + 2x + 3x + 4 + 12 = 70$ $6x + 16 = 70$

3. Subtract 16 from both sides: $6x = 54$

4. Divide both sides by 6: $x = 9$

So, the first odd integer is 9. Therefore, the three consecutive odd integers are:

• $x = 9$
• $x + 2 = 11$
• $x + 4 = 13$

Let's verify the solution:

• First integer: $9$
• Second integer: $11$
• Third integer: $13$
• Sum: $9 + 2 \cdot 11 + 3 \cdot 13 = 9 + 22 + 39 = 70$

The solution is correct. The three consecutive odd integers are 9, 11, and 13.

Would you like more details or have any questions?

Here are some related questions for further practice:

1. Find three consecutive even integers such that the sum of the first, twice the second, and three times the third is 108.
2. Find three consecutive integers such that the sum of the first, twice the second, and four times the third is 100.
3. Find three consecutive odd integers such that the sum of the first, three times the second, and five times the third is 140.
4. Find three consecutive even integers such that the sum of the first, four times the second, and five times the third is 180.
5. Find three consecutive odd integers such that the sum of the first, five times the second, and six times the third is 200.

Tip: When solving problems involving consecutive integers, remember to set up the equations correctly based on the given conditions. This will help in simplifying and solving the equations efficiently.