The 6th term of an arithmetic sequence is x while
the 11th is y find the 1st 2 terms
a+5d = x
a+10d = y
subtract them
5d = y-x
d = (y-x)/5
and
a + 5(y-x)/5 = x
a + y - x = x
a = 2x - y
first two terms are:
t1 = 2x-y
t2 = 2x-y + (y-x)/5 = (10x - 5y + y-x)/5
= (9x - 6y)/5
To find the first two terms of an arithmetic sequence with given terms, we need the values of the 6th (x) and 11th (y) terms. An arithmetic sequence is a sequence in which the difference between consecutive terms is constant.
Let's call the first term of the sequence "a" and the common difference between terms "d".
Now, we can use the formulas for the nth term of an arithmetic sequence:
The general formula for the nth term of an arithmetic sequence is:
An = a + (n - 1)d
Using this formula, we can find the values of the 6th and 11th terms:
For the 6th term:
A6 = a + (6 - 1)d = a + 5d = x ...(1)
For the 11th term:
A11 = a + (11 - 1)d = a + 10d = y ...(2)
We have two equations (equation 1 and 2) with two unknowns (a and d). We can solve these equations simultaneously to find the values of a and d.
First, let's subtract equation 1 from equation 2 to eliminate the 'a' term:
(A11 - A6) = (a + 10d) - (a + 5d)
Simplifying the equation:
y - x = 10d - 5d
y - x = 5d
Now, we can isolate 'd':
d = (y - x) / 5
Substituting this value of 'd' in equation 1, we can solve for 'a':
x = a + 5d
Substituting the value of 'd':
x = a + 5((y - x) / 5)
Simplifying the equation:
x = a + (y - x)
2x = a + y
Rearranging the equation:
a = 2x - y
Now that we have the values of 'a' and 'd', we can substitute them back into equation 1 to find the first term of the sequence:
x = a + 5d
Substituting the values of 'a' and 'd':
x = (2x - y) + 5((y - x) / 5)
Simplifying the equation:
x = 2x - y + (y - x)
x = x
Here, we get x = x, which tells us that there is no unique value for the first term 'a'. This means that there are many possible arithmetic sequences that satisfy the given conditions.
Without additional information, we cannot determine the specific values of the first two terms.