Consider an arithmetic which has the second term equal to 8 and the fifth term equal to 10
hmmm -- ver-r-r-ry nice!
a + d = 8
a + 4 d = 10
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0 - 3 d = -2
d = 2/3
a = 8 -2/3 = = 7 1/3 = 22/3
so
Tn = 22/3 + (n-1)* 2/3
To find the arithmetic sequence with the given information, we need to determine the common difference.
Let's denote the second term as a2 and the fifth term as a5.
Given:
a2 = 8
a5 = 10
We know that the formula for the nth term (an) of an arithmetic sequence is given by:
an = a1 + (n - 1)d
Where:
an is the nth term,
a1 is the first term,
n is the position of the term in the sequence,
d is the common difference.
Let's find the common difference (d) using the given information.
For the second term, a2, we can write:
a2 = a1 + (2 - 1)d
We are given that a2 = 8, so we have:
8 = a1 + d
For the fifth term, a5, we can write:
a5 = a1 + (5 - 1)d
We are given that a5 = 10, so we have:
10 = a1 + 4d
Now, we have a system of two equations with two variables (a1 and d):
1) 8 = a1 + d
2) 10 = a1 + 4d
We can solve this system by subtracting the first equation from the second equation:
10 - 8 = (a1 + 4d) - (a1 + d)
2 = 3d
Dividing both sides by 3, we get:
d = 2/3
Now, we can substitute the value of d back into either equation to find a1. Let's use the first equation:
8 = a1 + (2/3)
Multiply both sides by 3 to remove the fraction:
24 = 3a1 + 2
Subtract 2 from both sides:
22 = 3a1
Divide both sides by 3:
a1 = 22/3
So, the first term (a1) is 22/3 and the common difference (d) is 2/3.
Therefore, the arithmetic sequence is:
22/3, 8, 38/3, 14, ...
To find the common difference and the first term of the arithmetic sequence, we can use the formula for the nth term of an arithmetic sequence:
An = A1 + (n - 1) * d
where An represents the nth term, A1 is the first term, n is the term number, and d is the common difference.
Given that the second term (A2) is 8 and the fifth term (A5) is 10, we can set up two equations using the formula:
A2 = A1 + (2 - 1) * d
A5 = A1 + (5 - 1) * d
Substituting the known values:
8 = A1 + d
10 = A1 + 4d
Now we can solve this system of equations to find the values of A1 and d.
Subtracting the first equation from the second equation, we can eliminate A1:
10 - 8 = (A1 + 4d) - (A1 + d)
2 = 3d
Dividing both sides by 3:
d = 2/3
Substituting the value of d back into the first equation:
8 = A1 + (2/3)
Subtracting 2/3 from both sides:
A1 = 8 - (2/3)
Simplifying:
A1 = 22/3 or 7.33 (rounded to two decimal places)
Therefore, the first term (A1) in the arithmetic sequence is approximately 7.33 and the common difference (d) is 2/3.