The eighth term of an arithmetic series is twice the third term. The sum of the first eight terms are 39.Calculate the first term. Please help
the 8th term is 5 differences from the 3rd term, and 7 differences from the 1st term
e = 2 t = 2 (f + 2 d)
f + 7 d = 2 f + 4 d
... 3 d = f
sum8 = 8 f + 28 d = 39
substituting ... 24 d + 28 d = 39
... 52 d = 39 ... d = 3/4
substitute back to find f
To solve this problem, we need to use the formulas for the nth term and the sum of an arithmetic series.
1. Let's represent the first term of the arithmetic series as "a" and the common difference as "d".
2. The eighth term of the arithmetic series is given as "2 times the third term." So, the eighth term can be expressed as: a + 7d = 2(a + 2d).
3. Simplify and rearrange the equation: a + 7d = 2a + 4d.
Subtracting "a" and "4d" from both sides, we get: 7d - 4d = 2a - a,
which simplifies to: 3d = a.
4. The sum of the first eight terms of the arithmetic series is given as 39. The formula for the sum of an arithmetic series is: Sn = (n/2)(a + l), where Sn represents the sum of the first "n" terms, "a" is the first term, and "l" is the nth term.
5. Substituting the given values into the formula, we have: 39 = (8/2)(a + (a + 7d)).
Simplifying the equation further: 39 = 4(2a + 7d).
Divide both sides by 4: 9.75 = 2a + 7d.
6. Now, substitute the value of "a" (from step 3) into the equation: 9.75 = 2(3d) + 7d,
which simplifies to: 9.75 = 6d + 7d.
7. Combine like terms: 9.75 = 13d.
Divide both sides by 13: 9.75/13 = d.
The value of d is approximately 0.75.
8. Now, substitute the value of d into the equation from step 3: 3d = a.
Substituting the value of d, we get: 3(0.75) = a.
The value of a is 2.25.
Therefore, the first term of the arithmetic series is 2.25.
To calculate the first term of the arithmetic series, we first need to find the common difference (d). The common difference is the same amount added or subtracted to each term in the series.
Let's start by expressing the terms of the arithmetic series in terms of the first term (a) and the common difference (d).
The third term can be written as: a + 2d
The eighth term can be written as: a + 7d
We are given that the eighth term is twice the third term. So we can create an equation as follows:
a + 7d = 2(a + 2d)
Now, let's simplify the equation:
a + 7d = 2a + 4d
Rearranging the equation, we get:
7d - 4d = 2a - a
3d = a
Now, we have an expression for a in terms of d.
Next, we know that the sum of the first eight terms is 39. The sum of an arithmetic series can be calculated using the formula: S = (n/2)(2a + (n-1)d)
Plugging in the values, we have:
39 = (8/2)(2a + (8-1)d)
39 = 4(2a + 7d)
9.75 = 2a + 7d
Rearranging the equation again:
2a = 9.75 - 7d
Now, substitute the value of a from the earlier equation:
2(3d) = 9.75 - 7d
6d = 9.75 - 7d
13d = 9.75
Now, solve for d:
d = 9.75/13
d = 0.75
We now have the value of d.
Finally, substitute the value of d into the equation for a:
a = 3d
a = 3(0.75)
a = 2.25
Therefore, the first term of the arithmetic series is 2.25.