The fifth term of an arithmetic sequence is 23 and the 12th term is 72.
What is the value of the 10th term.
Which term has a value of 268.
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To find the value of the 10th term of the arithmetic sequence, we need to determine the common difference first.
Step 1: Find the common difference (d):
The difference between the fifth term (a₅) and the twelfth term (a₁₂) is given as:
a₁₂ - a₅ = 72 - 23 = 49
Since there are 7 terms between a₅ and a₁₂, the common difference (d) can be calculated as:
d = (a₁₂ - a₅) ÷ (12 - 5) = 49 ÷ 7 = 7
So, the common difference is 7.
Step 2: Find the value of the 10th term (a₁₀):
Now that we know the common difference (d), we can find the value of the 10th term (a₁₀) using the formula:
a₁₀ = a₁ + (n - 1) × d
Plugging in the values:
a₁₀ = a₅ + (10 - 5) × 7 = 23 + 5 × 7 = 23 + 35 = 58
Therefore, the value of the 10th term is 58.
To find which term has a value of 268, we will use the same formula.
Step 1: Find the value of n:
268 = a₁ + (n - 1) × d
Substituting the values we know:
268 = a₁ + (n - 1) × 7
We can simplify this equation.
Step 2: Solve for n:
Let's isolate n by subtracting a₁ from both sides of the equation:
268 - a₁ = (n - 1) × 7
Now we replace a₁ with its value of 23 that we initially had:
268 - 23 = (n - 1) × 7
245 = (n - 1) × 7
Next, we can divide both sides of the equation by 7 to solve for (n - 1):
35 = n - 1
Adding 1 to both sides gives us:
n = 36
So, the term with a value of 268 is the 36th term.
To find the value of the 10th term of the arithmetic sequence, we can use the formula for finding the nth term of an arithmetic sequence:
nth term = a + (n - 1) * d
Where:
- nth term represents the value of the term we want to find
- a represents the first term of the sequence
- n represents the position of the term we want to find
- d represents the common difference between consecutive terms of the sequence
Given that the fifth term is 23 and the twelfth term is 72, we can set up two equations to find the values of the first term (a) and the common difference (d).
Using the formula, we can set up the following two equations:
a + 4d = 23 ----(1) (since the fifth term is the first term + 4 times the common difference)
a + 11d = 72 ----(2) (since the twelfth term is the first term + 11 times the common difference)
To find the value of a and d, we can solve these two equations simultaneously. Subtracting equation (1) from equation (2), we get:
7d = 49
d = 49 / 7
d = 7
Now, substitute the value of d back into equation (1) or (2) to find the value of a.
a + 4 * 7 = 23
a + 28 = 23
a = 23 - 28
a = -5
So, the first term of the arithmetic sequence is -5, and the common difference is 7.
To find the value of the 10th term, substitute the values we have found into the formula:
10th term = -5 + (10 - 1) * 7
10th term = -5 + 9 * 7
10th term = -5 + 63
10th term = 58
Therefore, the value of the 10th term is 58.
To find which term has a value of 268, we can use the same formula:
nth term = a + (n - 1) * d
Substituting the known values:
268 = -5 + (n - 1) * 7
Simplifying the equation:
268 = -5 + 7n - 7
273 = 7n
n = 273 / 7
n = 39
So, the term that has a value of 268 is the 39th term.