The 4th term of an arithmetic sequence is 18 and the 6th term is 28. Give the first 3 terms. Use the formula
a+3d = 18
a+5d = 28
subtract them:
2d = 10
d = 5
a + 15 = 18
a = 3
now form the first 3 terms
Let's do some arithmetic, shall we? We know that the common difference (d) between terms in an arithmetic sequence is the same. So, we can find it by subtracting the fourth term from the sixth term: 28 - 18 = 10.
Now, we can use the formula for the nth term of an arithmetic sequence: an = a1 + (n - 1)d, where an is the nth term, a1 is the first term, n is the position of the term, and d is the common difference.
We have the value of the fourth term (18) and the common difference (10), so let's solve the equation for the first term (a1):
18 = a1 + (4 - 1) * 10
18 = a1 + 3 * 10
18 = a1 + 30
a1 = 18 - 30
a1 = -12
Now that we have the first term (a1 = -12), we can find the remaining terms:
Second term (a2) = a1 + (2 - 1) * d = -12 + 1 * 10 = -2
Third term (a3) = a1 + (3 - 1) * d = -12 + 2 * 10 = 8
So, the first three terms of the arithmetic sequence are -12, -2, and 8. I hope this arithmetic journey brought a smile to your face!
To find the first three terms of the arithmetic sequence, we can use the formula for the nth term of an arithmetic sequence:
a_n = a_1 + (n - 1)d
where:
a_n is the nth term,
a_1 is the first term, and
d is the common difference.
Given that the 4th term (a_4) is 18 and the 6th term (a_6) is 28, we can set up the following equations:
18 = a_1 + (4 - 1)d
28 = a_1 + (6 - 1)d
Simplifying the equations, we get:
18 = a_1 + 3d
28 = a_1 + 5d
Now, we can solve this system of equations to find the values of a_1 and d.
Subtracting the first equation from the second equation, we get:
10 = 2d
Dividing both sides by 2, we find:
d = 5
Substituting this value of d back into the first equation:
18 = a_1 + 3(5)
18 = a_1 + 15
a_1 = 18 - 15
a_1 = 3
So, the first term (a_1) is 3, and the common difference (d) is 5.
Now, we can find the first three terms using the formula.
a_1 = 3
a_2 = a_1 + d = 3 + 5 = 8
a_3 = a_2 + d = 8 + 5 = 13
Therefore, the first three terms of the arithmetic sequence are 3, 8, and 13.
To find the first three terms of an arithmetic sequence, we can use the formula for the nth term of an arithmetic sequence:
\[a_n = a_1 + (n - 1) \cdot d\]
where \(a_n\) represents the nth term, \(a_1\) represents the first term, \(n\) represents the position of the term, and \(d\) represents the common difference between consecutive terms.
Given that the 4th term is 18 and the 6th term is 28, we can substitute these values into the formula to form two equations:
For the 4th term:
\[18 = a_1 + (4 - 1) \cdot d\]
For the 6th term:
\[28 = a_1 + (6 - 1) \cdot d\]
Since we want to find the first three terms, we can solve these two equations to determine the values of \(a_1\) and \(d\).