what are the four arithmetic mean between 8 and 18 solution po show solution
The correct answer and explanation is:
To find the four arithmetic means between 8 and 18, we are essentially inserting four numbers between 8 and 18 such that the entire sequence forms an arithmetic sequence (a sequence with a constant difference between each term).
✅ Step-by-Step Solution:
We want to find four arithmetic means between 8 and 18.
That means the sequence will have:
8, ___, ___, ___, ___, 18
This is a total of 6 terms.
Let’s denote this arithmetic sequence as:
a₁, a₂, a₃, a₄, a₅, a₆
Where:
- a₁ = 8
- a₆ = 18
- Total terms (n) = 6
- Common difference = d
👉 Step 1: Use the formula for the nth term of an arithmetic sequence:
an=a1+(n−1)⋅da_n = a_1 + (n – 1) \cdot d
We know: a6=8+(6−1)d=18a_6 = 8 + (6 – 1)d = 18 8+5d=188 + 5d = 18 5d=105d = 10 d=2d = 2
👉 Step 2: Use the common difference to find the missing terms:
Now that we know the common difference is 2:
- a₁ = 8
- a₂ = 8 + 2 = 10
- a₃ = 10 + 2 = 12
- a₄ = 12 + 2 = 14
- a₅ = 14 + 2 = 16
- a₆ = 16 + 2 = 18 ✅
✅ Final Answer:
The four arithmetic means between 8 and 18 are:
10, 12, 14, 16
🧠 300-Word Explanation:
In mathematics, an arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This difference is called the common difference (d).
To find multiple arithmetic means between two numbers, we essentially construct a longer arithmetic sequence that starts with the first number and ends with the last, and the terms in between are the arithmetic means we’re looking for.
In this case, we need four arithmetic means between 8 and 18. This means our full sequence must have six terms in total (including the given first and last terms).
We use the arithmetic formula for the nth term: an=a1+(n−1)da_n = a_1 + (n – 1)d
By plugging in the values (with a₆ = 18 and a₁ = 8), we can solve for the common difference, which turns out to be 2.
Then, using this common difference, we calculate the four missing values between 8 and 18 by repeatedly adding 2. This gives us 10, 12, 14, and 16 as the four arithmetic means.
This approach ensures the sequence increases uniformly, maintaining the essential property of an arithmetic sequence.