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 i've been stuck for hours on this question
In that case, you must have forgotten your Algebra I. Look at what you know:
a+7d = 2(a+2d)
8/2 (2a+7d) = 39
or, massaged a bit,
a = 3d
8a+28d = 39
To solve this problem, we can use the formula for the nth term of an arithmetic series:
\$a_n = a_1 + (n-1)d\$
Where:
- \$a_n\$ is the nth term of the series
- \$a_1\$ is the first term of the series
- \$d\$ is the common difference between consecutive terms
- \$n\$ is the number of terms
Let's break down the information given step by step:
1. The eighth term (\$a_8\$) is twice the third term (\$a_3\$).
=> \$a_8 = 2 \cdot a_3\$
2. The sum of the first eight terms is 39.
=> \$a_1 + a_2 + a_3 + \ldots + a_8 = 39\$
Now, let's solve for the first term (\$a_1\$).
We'll start with the equation for the sum of the eight terms and substitute the values of \$a_1\$, \$a_3\$, and \$a_8\$:
\$a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7 + a_8 = 39\$
Since \$a_2 = a_1 + d\$, \$a_3 = a_1 + 2d\$, and \$a_8 = a_1 + 7d\$, we can rewrite the equation as:
\$a_1 + (a_1 + d) + (a_1 + 2d) + \ldots + (a_1 + 7d) = 39\$
Now, substitute the equation for \$a_8\$:
\$a_1 + (a_1 + d) + (a_1 + 2d) + \ldots + (a_1 + 6d) + 2(a_1 + 2d) = 39\$
Simplify the equation:
\$8a_1 + 28d = 39\$
We also have the equation given in the problem that the eighth term is twice the third term:
\$a_8 = 2 \cdot a_3\$
Substitute the values for \$a_8\$ and \$a_3\$:
\$a_1 + 7d = 2(a_1 + 2d)\$
Simplify the equation:
\$a_1 + 7d = 2a_1 + 4d\$
\$3d = a_1\$
Now we have two equations:
1. \$8a_1 + 28d = 39\$
2. \$3d = a_1\$
Using equation 2, substitute \$a_1\$ with \$3d\$ in equation 1:
\$8(3d) + 28d = 39\$
Simplify and solve for \$d\$:
\$24d + 28d = 39\$
\$52d = 39\$
\$d = \frac{39}{52}\$
Now that we know the value of \$d\$, we can find the value of \$a_1\$ by substituting it into equation 2:
\$3(\frac{39}{52}) = a_1\$
Simplify:
\$a_1 = \frac{117}{52}\$
Therefore, the first term is \$\frac{117}{52}\$ or approximately 2.25.