The fifth term of a linear sequence is 17 and the third term is 11, Find the sum of the first seven terms
3rd to 5th is 2r distance, so r=1/2 (17-11)=3
given 5th , work back
17,14,11,8,5 so 5 is the first term.
20,23 are the remaining terms of the first seven, add them.
This type of sequence is also called arithmetic sequence, and there are formulas associated with it, and you ought to review them.
Pardon
To find the sum of the first seven terms, we need to determine the common difference (d) of the sequence.
We know that the fifth term (a5) is 17, and the third term (a3) is 11.
Using the formula for the nth term of a linear sequence:
an = a1 + (n - 1)d,
we can set up two equations using the given information.
1) a5 = a1 + 4d = 17
2) a3 = a1 + 2d = 11
Now, we can solve these equations to determine the values of a1 and d.
Subtracting equation 2) from equation 1), we get:
a5 - a3 = (a1 + 4d) - (a1 + 2d)
17 - 11 = 2d
6 = 2d
Dividing both sides by 2, we get:
d = 3
Substituting this value back into equation 2), we can find a1:
a1 + 2d = 11
a1 + 2(3) = 11
a1 + 6 = 11
a1 = 11 - 6
a1 = 5
So, the first term (a1) is 5, and the common difference (d) is 3.
To find the sum of the first seven terms (S7), we can use the formula for the sum of an arithmetic series:
S7 = (n/2)(2a1 + (n - 1)d),
where n is the number of terms.
Plugging in the values:
S7 = (7/2)(2(5) + (7 - 1)(3))
S7 = (7/2)(10 + 6(3))
S7 = (7/2)(10 + 18)
S7 = (7/2)(28)
S7 = 7(14)
S7 = 98
Therefore, the sum of the first seven terms of this linear sequence is 98.
To find the sum of the first seven terms of a linear sequence, we first need to find the common difference of the sequence.
A linear sequence is characterized by a constant difference between consecutive terms. Let's denote the first term of the sequence as "a" and the common difference as "d."
We are given that the third term of the sequence is 11, which can be written as:
a + 2d = 11 ----(1)
We are also given that the fifth term of the sequence is 17:
a + 4d = 17 ----(2)
To solve this system of equations, we can subtract equation (1) from equation (2):
(a + 4d) - (a + 2d) = 17 - 11
Simplifying, we get:
2d = 6
Dividing both sides by 2:
d = 3
Now that we have the common difference, we can find the first term "a" of the sequence.
Using equation (1), we substitute the value of d = 3:
a + 2(3) = 11
Simplifying, we get:
a + 6 = 11
Subtracting 6 from both sides:
a = 5
Therefore, the first term of the sequence is a = 5 and the common difference is d = 3.
Now, we can find the sum of the first seven terms of the sequence using the formula for the sum of an arithmetic series:
Sn = (n/2)(2a + (n-1)d)
where Sn represents the sum of the first n terms, a is the first term, and d is the common difference.
Substituting the given values:
S7 = (7/2)(2(5) + (7-1)(3))
Simplifying, we get:
S7 = (7/2)(10 + 6(3))
S7 = (7/2)(10 + 18)
S7 = (7/2)(28)
S7 = 7 * 14
S7 = 98
Therefore, the sum of the first seven terms of the linear sequence is 98.