The 7th and 13th term of an arithmetic sequence are 51 an 15 respectively. which term of the sequence is equal to -21
the 13th term is six terms away from the 7th, so
7d = 15-51 = -36
Now you know that the terms drop by 36 every 7 terms.
Since -21 = 15-36, there are 7 more terms, so the 20th term is -21
To find the term of the sequence that is equal to -21, we first need to find the common difference (d) of the arithmetic sequence.
The formula for the nth term of an arithmetic sequence is:
an = a1 + (n - 1)d
Where:
- an is the nth term of the sequence
- a1 is the first term of the sequence
- d is the common difference between terms
- n is the position of the term in the sequence
We are given the values of the 7th term (a7) and the 13th term (a13), which are 51 and 15, respectively. Using this information, we can set up two equations:
a7 = a1 + (7 - 1)d
51 = a1 + 6d
a13 = a1 + (13 - 1)d
15 = a1 + 12d
Now we have a system of two equations with two variables (a1 and d). To solve this system, we can use the method of substitution or elimination.
Let's use the method of elimination to solve the system:
Multiply the first equation by 2:
102 = 2a1 + 12d
Subtract the second equation from the multiplied first equation:
102 - 15 = 2a1 + 12d - (a1 + 12d)
87 = a1
Now that we have the value of a1, we can substitute it into one of the original equations to find the common difference (d). Let's use the first equation:
51 = 87 + 6d
Subtract 87 from both sides:
51 - 87 = 6d
-36 = 6d
Divide both sides by 6:
-6 = d
So the common difference (d) is -6.
Now that we know the value of d, we can find the term of the sequence that is equal to -21 by substituting it into the formula for the nth term:
an = a1 + (n - 1)d
-21 = 87 + (n - 1)(-6)
Rearrange the equation:
-21 = 87 - 6(n - 1)
Combine like terms:
-21 = 87 - 6n + 6
-21 = 93 - 6n
Subtract 93 from both sides:
-21 - 93 = -6n
-114 = -6n
Divide both sides by -6:
19 = n
Therefore, the term of the sequence that is equal to -21 is the 19th term.