The difference between the second and fourth terms of an arithmetic sequence is 3. If the product of the first and fourth term is 34, calculate the value of the first term.
To find the value of the first term in the arithmetic sequence, we need to use the given information.
Let's first assign variables to the terms in the arithmetic sequence. We'll let "a" represent the first term, and "d" represent the common difference between the terms.
The second term in the sequence can be expressed as "a + d". The fourth term can be expressed as "a + 3d" because there is a common difference of "d" between each term.
Now we are given that the difference between the second and fourth terms is 3. So we can write the equation:
(a + 3d) - (a + d) = 3
By simplifying the equation, we get:
a + 3d - a - d = 3
2d = 3
Now let's address the second condition, where the product of the first and fourth terms is 34. We can write this as:
a * (a + 3d) = 34
Expanding and simplifying the equation, we have:
a^2 + 3ad = 34
Now we have a system of two equations:
2d = 3 (Equation 1)
a^2 + 3ad = 34 (Equation 2)
From equation 1, we can solve for d:
2d = 3
d = 3/2
Now we substitute the value of d into equation 2:
a^2 + 3ad = 34
a^2 + 3a(3/2) = 34
a^2 + (9/2)a = 34
2a^2 + 9a - 68 = 0
We have a quadratic equation. To solve for "a", we can factor the equation or use the quadratic formula. By factoring, we can find that:
(a + 17)(2a - 4) = 0
Setting each factor equal to zero gives us two possible solutions:
a + 17 = 0 (reject, as a cannot be negative)
or
2a - 4 = 0
2a = 4
a = 2
Therefore, the first term in the arithmetic sequence is 2.
clearly, d = 3/2.
So, if the first term is a,
a(a+3d) = 34
a(a+9/2) = 34
. . .