The 8th term of a linear sequence is 18 and the 12th term is 26.find the first. The common difference the 20th term
In this sequence:
a8 = a1 + 7 d = 18 , a12 = a1 + 11 d = 26
Now solve system:
a1 + 7 d = 18
a1 + 11 d = 26
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a1 + 7 d = 18
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a1 + 11 d = 26
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a1 - a1 + 7 d - 11 d = 18 - 26
0 - 4 d = - 8
- 4 d = - 8
d = - 8 / - 4 = 2
a1 + 7 d = 18
a1 + 7 ∙ 2 = 18
a1 + 14 = 18
a1 = 18 - 14 = 4
a20 = a1 + 19 d = 4 + 19 ∙ 2 = 4 + 38 = 42
Please 8th by 26 term
To find the first term and the common difference of a linear sequence, we can use the formula for the nth term of an arithmetic sequence:
nth term = first term + (n - 1) * common difference
Let's use this formula to solve the problem step-by-step.
Step 1: Calculate the common difference (d) using the information given.
12th term = 26
8th term = 18
Using the formula, we can write two equations:
18 = first term + (8 - 1) * d (Equation 1)
26 = first term + (12 - 1) * d (Equation 2)
Simplify these equations:
18 = first term + 7d
26 = first term + 11d
Step 2: Solve the system of equations.
We can solve these equations by subtracting Equation 1 from Equation 2:
26 - 18 = (first term + 11d) - (first term + 7d)
8 = 4d
Divide both sides by 4:
d = 2
Step 3: Calculate the first term (a) using the common difference (d).
From Equation 1, we can substitute the value of d:
18 = first term + 7 * 2
Simplify the equation:
18 = first term + 14
Subtract 14 from both sides:
4 = first term
So, the first term is 4.
Step 4: Calculate the 20th term using the first term and the common difference.
20th term = first term + (20 - 1) * common difference
= 4 + 19 * 2
= 4 + 38
= 42
So, the 20th term is 42.
To find the first term and the common difference of a linear sequence, we can use the formula for the general term of an arithmetic sequence. The formula is:
an = a1 + (n-1)d
where:
an represents the nth term of the sequence,
a1 represents the first term of the sequence, and
d represents the common difference between consecutive terms.
We are given two pieces of information:
1. The 8th term, a8, is 18: a8 = 18
2. The 12th term, a12, is 26: a12 = 26
Using the formula, we can substitute these values and create two equations:
a8 = a1 + (8-1)d -- Equation 1
a12 = a1 + (12-1)d -- Equation 2
Substituting the given information into the equations, we get:
18 = a1 + 7d -- Equation 1
26 = a1 + 11d -- Equation 2
Now we have a system of two equations with two variables. To solve it, we can use either substitution or elimination method.
Let's solve it using the elimination method:
Multiply Equation 1 by 11 and Equation 2 by 7 to eliminate the variable 'a1':
198 = 11a1 + 77d -- Equation 3
182 = 7a1 + 77d -- Equation 4
Subtract Equation 4 from Equation 3 to eliminate 'a1':
16 = 4a1
Divide both sides of the equation by 4:
4 = a1
Now we have found the value of the first term, which is 4.
To find the common difference, we can substitute the value of 'a1' back into one of the original equations, such as Equation 1:
18 = 4 + 7d
Subtract 4 from both sides:
14 = 7d
Divide both sides by 7:
2 = d
So, the first term is 4, and the common difference is 2.
Finally, to find the 20th term, we can substitute the values of 'a1' and 'd' into the formula for the general term of an arithmetic sequence:
a20 = a1 + (20-1)d
Substituting the values, we get:
a20 = 4 + (20-1)2
a20 = 4 + 19*2
a20 = 4 + 38
a20 = 42
Therefore, the 20th term of the sequence is 42.