the third and fourth terms of the sequence are 21 and 38. If the second differences are a constant 2, what are the first five terms of the sequence?
The 1st difference between terms 3 and 4 is 17
The differences and the sequence are thus
2 2 2 2 2 2 2 ...
13 15 17 19 21 23 ...
-7 6 21 38 57 78 101 ...
The first term of a sequence is 8. The first of the first differences is 3. The second differences are a constant 1. What are the first five terms of the sequence?
To find the first five terms of the sequence, we'll have to analyze the given information and use it to determine the pattern.
Let's call the first term of the sequence "a" and the common difference between consecutive terms "d." With this information, we can create an equation for each term.
The third term of the sequence can be expressed as:
a + (2-1)*d = a + d
The fourth term of the sequence can be expressed as:
a + (3-1)*d = a + 2d
Given that the third term is 21 and the fourth term is 38, we can write the following equations:
a + d = 21 (Equation 1)
a + 2d = 38 (Equation 2)
Now, let's solve this system of equations to find the values of "a" and "d":
Subtract Equation 1 from Equation 2:
(a + 2d) - (a + d) = 38 - 21
d = 17
Substitute the value of "d" back into Equation 1 to solve for "a":
a + 17 = 21
a = 4
Therefore, the first term of the sequence is 4, and the common difference is 17.
Now that we have the first term and the common difference, we can easily find the subsequent terms of the sequence. The general formula for the nth term of an arithmetic sequence is:
tn = a + (n-1)d
Using this formula, we can find the first five terms of the sequence:
t1 = 4 + (1-1)*17 = 4
t2 = 4 + (2-1)*17 = 21
t3 = 4 + (3-1)*17 = 38
t4 = 4 + (4-1)*17 = 55
t5 = 4 + (5-1)*17 = 72
Therefore, the first five terms of the sequence are: 4, 21, 38, 55, 72.