The sum of the third and seventh terms of a linear sequence is 38 and the ninth term is 37.find the sequence.
just use your definitions
e.g.
term(3) = a + 2d , etc
a+2d + a+6d = 38
2a + 8d = 38
a + 4d = 19 ***
a + 8d = 37 **
subtract them:
4 d = 18
d = 9/2 or 4.5
into **
a + 72/2 = 37
a + 36 = 37
a = 1
the sequence is:
1 , 5.5, 10, 14.5, 19, 23.5, 28, ...
Let's consider the linear sequence as follows: a, a + d, a + 2d, a + 3d, ...
We know that the third term is a + 2d and the seventh term is a + 6d. The sum of these terms is given as 38:
(a + 2d) + (a + 6d) = 38
Simplifying this equation, we get:
2a + 8d = 38
Similarly, the ninth term is a + 8d, and it is given that the ninth term is 37:
a + 8d = 37
Now, we have a system of equations:
2a + 8d = 38
a + 8d = 37
To solve this system of equations, we can use the method of substitution.
From the second equation, we can solve for a:
a = 37 - 8d
Substituting this value of a into the first equation:
2(37 - 8d) + 8d = 38
74 - 16d + 8d = 38
74 - 8d = 38
-8d = 38 - 74
-8d = -36
d = -36 / (-8)
d = 4.5
Now, substituting the value of d back into the second equation to find a:
a + 8(4.5) = 37
a + 36 = 37
a = 37 - 36
a = 1
So, the first term of the sequence is 1 and the common difference is 4.5.
Thus, the sequence is: 1, 5.5, 10, 14.5, 19, 23.5, 28, 32.5, 37.
To find the sequence, let's denote the first term as 'a' and the common difference between terms as 'd'.
In a linear sequence, the formula to calculate the nth term is given by:
Tn = a + (n - 1) * d
Using this formula, we can write the equations based on the given information:
Equation 1: T3 + T7 = 38
Equation 2: T9 = 37
Substituting the values using Equation 1:
(a + 2d) + (a + 6d) = 38
2a + 8d = 38
Simplifying Equation 2:
a + 8d = 37
Now, we have a system of two equations with two variables (a and d):
2a + 8d = 38 (Equation 1)
a + 8d = 37 (Equation 2)
We can solve this system of equations using various methods like substitution or elimination:
Subtracting Equation 2 from Equation 1:
(2a + 8d) - (a + 8d) = 38 - 37
a = 1
Substituting this value back into Equation 2:
1 + 8d = 37
8d = 36
d = 4
Therefore, the first term (a) is 1, and the common difference (d) is 4. Now, we can calculate the sequence:
T1 = a = 1
T2 = a + d = 1 + 4 = 5
T3 = a + 2d = 1 + 2(4) = 9
T4 = a + 3d = 1 + 3(4) = 13
T5 = a + 4d = 1 + 4(4) = 17
T6 = a + 5d = 1 + 5(4) = 21
T7 = a + 6d = 1 + 6(4) = 25
T8 = a + 7d = 1 + 7(4) = 29
T9 = a + 8d = 1 + 8(4) = 33
Therefore, the sequence is: 1, 5, 9, 13, 17, 21, 25, 29, 33.