If the 15th and 16th term of an arithmetic sequence are 99 and 92 respectively then the 5th term is?
Studying for math final and I cannot figure this out
To find the 5th term of an arithmetic sequence, we need to determine the common difference between the terms first.
The formula to find any term in an arithmetic sequence is given by:
\[ a_n = a_1 + (n-1)d \]
Where:
\( a_n \) = the \( n \)th term
\( a_1 \) = the first term
\( n \) = the term number
\( d \) = the common difference
Given:
\( a_{15} = 99 \)
\( a_{16} = 92 \)
Let's substitute these values into the formula to find the common difference:
For \( a_{15} = 99 \):
\[ a_{15} = a_1 + (15-1)d \]
\[ 99 = a_1 + 14d \]
For \( a_{16} = 92 \):
\[ a_{16} = a_1 + (16-1)d \]
\[ 92 = a_1 + 15d \]
Now, we have a system of two equations:
Equation 1: \( 99 = a_1 + 14d \)
Equation 2: \( 92 = a_1 + 15d \)
Subtracting Equation 2 from Equation 1 will eliminate \( a_1 \):
\[ (99 - 92) = (a_1 + 14d) - (a_1 + 15d) \]
\[ 7 = -d \]
\[ d = -7 \]
Now that we have the common difference, we can find the 5th term of the arithmetic sequence.
Using the formula again:
\[ a_n = a_1 + (n-1)d \]
For \( a_5 \):
\[ a_5 = a_1 + (5-1)(-7) \]
\[ a_5 = a_1 - 28 \]
Unfortunately, we don't have enough information to directly find the value of \( a_1 \).
To find the 5th term of an arithmetic sequence, we need to know the common difference between terms.
Given that the 15th and 16th terms of the arithmetic sequence are 99 and 92 respectively, we can find the common difference by subtracting the 15th term from the 16th term:
92 - 99 = -7
Now that we know the common difference, we can proceed to find the 5th term.
We know that the 15th term is 99, which means that we are going back 10 terms (since 15 - 5 = 10).
To find the 5th term, we can subtract the common difference multiplied by 10 from the 15th term:
99 - (-7 * 10) = 99 + 70 = 169
Therefore, the 5th term of the arithmetic sequence is 169.
the difference is -7
the 5th term is 10 differences before the 15th term