The sixth term of an arithmetic sequence is 20.6 and the 9th term is 30.2. Find the 20th term and find the nth term.
In arithmetic sequence :
an = a1 + ( n - 1 ) * d
d is the common difference
n is the number of the term to find
a6 = a1 + ( 6 - 1 ) * d = 20.6
a1 + 5 d = 20.6
a9 = a1 + ( 9 - 1 ) * d = 30.2
a1 + 8 d = 30.2
Now you must solve system of two equations with two unknown :
a1 + 5 d = 20.6
a1 + 8 d = 30.2
The solutions are :
a1 = 23 / 5 = 4.6
d = 16 / 5 = 3.2
a20 = a1 + ( n - 1 ) * d
a20 = 4.6 + ( 20 - 1 ) * 3.2
a20 = 4.6 + 19 * 3.2
a20 = 4.6 + 60.8
a20 = 65.4
an = a1 + ( n - 1 ) * d
an = 4.6 + ( n - 1 ) * 3.2
In an arithmetic sequence,
an = a1 + (n-1)d
where
an = nth term
a1 = first term
n = number of terms
d = common difference
The sixth term is 20.6 so,
20.6 = a1 + (6-1)d
20.6 = a1 + 5d
The ninth term is 30.2 so,
30.2 = a1 + (9-1)d
30.2 = a1 + 8d
Now you have two equations, to unknowns. We can subtract them to solve for d:
20.6 = a1 + 5d
-(30.2 = a1 + 8d)
--------------------------
-9.6 = -3d
d = 3.2
And thus,
20.6 = a1 + 5(3.2)
a1 = 4.6
Now you have values for a1 and d, you can solve for the 20th term.
To find the 20th term of an arithmetic sequence, we can use the formula:
\[ a_n = a_1 + (n-1)d \]
where \( a_n \) represents the \( n \)-th term, \( a_1 \) represents the first term, \( n \) represents the position of the term, and \( d \) represents the common difference between the terms.
We are given that the 6th term of the sequence is 20.6, so we can substitute \( a_1 = 20.6 \) and \( n = 6 \) into the formula:
\[ 20.6 = a_1 + (6-1)d \]
Simplifying the equation, we get:
\[ 20.6 = a_1 + 5d \]
Similarly, we are given that the 9th term is 30.2, so we can substitute \( a_1 = 30.2 \) and \( n = 9 \) into the formula:
\[ 30.2 = a_1 + (9-1)d \]
Simplifying the equation, we get:
\[ 30.2 = a_1 + 8d \]
Now we have a system of two equations with two variables ( \( a_1 \) and \( d \) ). We can solve this system to find the values of \( a_1 \) and \( d \) using any method, such as substitution or elimination.
Subtracting the first equation from the second equation, we get:
\[ 30.2 - 20.6 = (a_1 + 8d) - (a_1 + 5d) \]
Simplifying the equation, we get:
\[ 9.6 = 3d \]
Dividing both sides of the equation by 3, we get:
\[ d = 3.2 \]
Now we can substitute this value of \( d \) into either of the original equations to solve for \( a_1 \).
Using the first equation:
\[ 20.6 = a_1 + 5(3.2) \]
Simplifying the equation, we get:
\[ 20.6 = a_1 + 16 \]
Subtracting 16 from both sides of the equation, we get:
\[ a_1 = 20.6 - 16 \]
\[ a_1 = 4.6 \]
So, the first term of the sequence (\( a_1 \)) is 4.6 and the common difference (\( d \)) is 3.2.
To find the 20th term of the sequence, we can substitute the values \( a_1 = 4.6 \), \( n = 20 \), and \( d = 3.2 \) into the formula:
\[ a_{20} = 4.6 + (20-1)3.2 \]
Simplifying the equation, we get:
\[ a_{20} = 4.6 + 19(3.2) \]
\[ a_{20} = 4.6 + 60.8 \]
\[ a_{20} = 65.4 \]
Therefore, the 20th term of the arithmetic sequence is 65.4.
To find the \( n \)-th term of the sequence, we can simply use the formula:
\[ a_n = a_1 + (n-1)d \]
Substituting \( a_1 = 4.6 \) and \( d = 3.2 \) into the formula, we get:
\[ a_n = 4.6 + (n-1)3.2 \]
So, the \( n \)-th term of the arithmetic sequence is \( a_n = 4.6 + (n-1)3.2 \).