The following problem refers to an arithmetic sequence. If the seventh term is 9 and the fourteenth term is 16, find the term a1, the common difference
d, and then find a27 and S27
i am struck please help me....
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To find the first term (a1) and the common difference (d) of an arithmetic sequence, you can use the formulas:
a1 = a - (n - 1) × d (1)
an = a + (n - 1) × d (2)
where a is the first term, an is the nth term, n is the position of the term in the sequence, and d is the common difference.
1. Finding a1 and d:
Given that the seventh term (a7) is 9 and the fourteenth term (a14) is 16, we can substitute these values into equation (2) to form two equations:
a7 = a + (7 - 1) × d (3)
a14 = a + (14 - 1) × d (4)
Substituting 9 for a7 and 16 for a14:
9 = a + 6d (5)
16 = a + 13d (6)
Now, we have a system of two equations (equations 5 and 6) with two variables (a and d).
To solve this system, we can use any method, such as substitution or elimination.
Using the method of elimination, let's multiply equation (5) by 13 and equation (6) by 6 to eliminate the a term:
117 = 13a + 78d (7)
96 = 6a + 78d (8)
Subtracting equation (8) from equation (7):
21 = 7a
Dividing both sides by 7:
a = 3
Substituting the value of a into equation (5):
9 = 3 + 6d
Subtracting 3 from both sides:
6 = 6d
Dividing both sides by 6:
d = 1
Therefore, the first term (a1) is 3, and the common difference (d) is 1.
2. Finding a27:
To find the 27th term (a27) in the arithmetic sequence, we can use equation (2):
an = a + (n - 1) × d
Substituting the values we found:
a27 = 3 + (27 - 1) × 1
Simplifying:
a27 = 3 + 26
a27 = 29
Therefore, the 27th term (a27) in the sequence is 29.
3. Finding S27:
To find the sum of the first 27 terms (S27) of an arithmetic sequence, we can use the formula for the sum of an arithmetic series:
Sn = n / 2 × (2a + (n - 1) × d)
Substituting the values we found:
S27 = 27 / 2 × (2 × 3 + (27 - 1) × 1)
Simplifying:
S27 = 13.5 × (6 + 26)
S27 = 13.5 × 32
S27 = 432
Therefore, the sum of the first 27 terms (S27) of the arithmetic sequence is 432.
I hope this explanation helps you understand how to solve the problem!