To find the sum of five arithmetic means between 34 and 42, follow these steps:

Step 1: Determine the number of terms in the sequence

There are five arithmetic means between 34 and 42. Including 34 and 42, the total number of terms $n$ in the sequence is: $n = 5 + 2 = 7$

Step 2: Determine the common difference $d$

The general formula for the $n$-th term of an arithmetic sequence is: $a_n = a_1 + (n-1)d$ Here:

• $a_1 = 34$ (the first term)
• $a_7 = 42$ (the seventh term)

Substitute the known values into the formula for the seventh term: $42 = 34 + (7-1)d$ $42 = 34 + 6d$ $6d = 42 - 34 = 8$ $d = \frac{8}{6} = \frac{4}{3} \approx 1.33$ So, the common difference $d$ is $\frac{4}{3}$.

Step 3: Determine all the terms

Now that we have the common difference $d = \frac{4}{3}$, we can find all the terms of the sequence:

• First term $a_1 = 34$
• Second term $a_2 = 34 + \frac{4}{3} = \frac{102}{3} + \frac{4}{3} = \frac{106}{3} \approx 35.33$
• Third term $a_3 = 35.33 + \frac{4}{3} = \frac{110}{3} \approx 36.67$
• Fourth term $a_4 = 36.67 + \frac{4}{3} = \frac{114}{3} \approx 38$
• Fifth term $a_5 = 38 + \frac{4}{3} = \frac{118}{3} \approx 39.33$
• Sixth term $a_6 = 39.33 + \frac{4}{3} = \frac{122}{3} \approx 40.67$
• Seventh term $a_7 = 42$

So, the sequence is approximately: $34, 35.33, 36.67, 38, 39.33, 40.67, 42$

Step 4: Calculate the sum of the sequence

The sum $S_n$ of the arithmetic sequence can be calculated using the formula: $S_n = \frac{n}{2} \times (a_1 + a_n)$ Substituting the known values: $S_7 = \frac{7}{2} \times (34 + 42) = \frac{7}{2} \times 76 = \frac{532}{2} = 266$

Final Answer

The sum of all the terms in the sequence is 266.

Would you like further details or have any questions about this? Here are five related questions you might find interesting:

1. How does the sum of an arithmetic sequence change if the number of terms is increased?
2. How do you derive the formula for the sum of an arithmetic series?
3. What happens to the common difference if the first and last terms remain the same but the number of terms increases?
4. How can you verify the arithmetic sequence if the common difference is a fraction?
5. How would you find the sum if the arithmetic means were all negative?

Tip: When working with arithmetic sequences, especially with fractional common differences, ensure each step is carefully calculated to avoid errors in determining the sequence terms.