The third term of an arithmetic sequence is -12 and the seventh term is 8.what is the sum of the first 10 terms?
Use your definitions and formulas
a + 2d = -12
a + 6d = 8
subtract the two equations, and find d
once you have d, go back into the first equation to find a
then find a + 9d
Well, it looks like the arithmetic sequence has quite a funny plot twist. The third term is -12, trying to be all negative and gloomy, but then the seventh term comes in all positive and chirpy with a value of 8. I must say, I didn't see that coming!
To find the sum of the first 10 terms, we can use the formula for the sum of an arithmetic sequence: S = (n/2)(2a + (n-1)d), where S is the sum, a is the first term, d is the common difference, and n is the number of terms.
Given that the third term is -12, we can find the common difference (d). Similarly, given that the seventh term is 8, we can find the first term (a).
Let's start with finding d: d = (term 7 - term 3) / (7 - 3). Plugging in the values, we get d = (8 - (-12)) / 4 = 5.
Now let's find a: a = term 3 - (3 - 1) * d. Substituting the values, we have a = -12 - (2 * 5) = -22.
Now that we have a and d, we can use the formula to find the sum (S) of the first 10 terms: S = (10/2)(2 * -22 + (10-1) * 5).
Calculating that out, we get S = 5 * (-44 + 9 * 5) = 5 * (-44 + 45) = 5 * 1 = 5.
So, the sum of the first 10 terms is 5. It seems like our arithmetic sequence really knows how to play tricks on us with its funny values!
To find the sum of the first 10 terms of an arithmetic sequence, we first need to determine the common difference (d) of the sequence.
Given that the third term (a3) is -12 and the seventh term (a7) is 8, we can use this information to find the common difference:
d = a7 - a3
d = 8 - (-12)
d = 8 + 12
d = 20
Now that we know the common difference (d = 20), we can find the first term (a1) of the arithmetic sequence.
a1 = a7 - (n - 1) * d
a1 = 8 - (7 - 1) * 20
a1 = 8 - 6 * 20
a1 = 8 - 120
a1 = -112
Using the first term (a1 = -112), the common difference (d = 20), and the number of terms (n = 10), we can find the sum of the first 10 terms using the formula:
Sum = n/2 * (2a1 + (n - 1) * d)
Sum = 10/2 * (2 * -112 + (10 - 1) * 20)
Sum = 5 * (-224 + 9 * 20)
Sum = 5 * (-224 + 180)
Sum = 5 * (-44)
Sum = -220
Therefore, the sum of the first 10 terms of the arithmetic sequence is -220.
To find the sum of the first 10 terms of an arithmetic sequence, we need to use the formula for the nth term of an arithmetic sequence and the formula for the sum of an arithmetic sequence.
An arithmetic sequence is a sequence in which the difference between any two consecutive terms is constant. The formula for the nth term of an arithmetic sequence is given by:
an = a1 + (n-1)d
where:
an represents the nth term,
a1 represents the first term,
n represents the term number, and
d represents the common difference.
In this case, we are given that the third term is -12, so a3 = -12, and the seventh term is 8, so a7 = 8. We need to find the value of a1 (first term) and d (common difference).
Using the formula for an arithmetic sequence, we can set up two equations using the values given:
a3 = a1 + (3-1)d (equation 1)
a7 = a1 + (7-1)d (equation 2)
Substitute the given values:
-12 = a1 + 2d (equation 1)
8 = a1 + 6d (equation 2)
Solving these equations will give us the values of a1 and d. Subtracting equation 1 from equation 2 will eliminate a1:
8 - (-12) = (a1 + 6d) - (a1 + 2d)
20 = 4d
Dividing both sides of the equation by 4, we get:
d = 5
Substituting the value of d back into either equation, we can solve for a1:
-12 = a1 + 2(5)
-12 = a1 + 10
Subtract 10 from both sides:
-22 = a1
So, the value of a1 is -22 and the common difference (d) is 5.
Now, we can use the formula to find the sum of the first 10 terms of the arithmetic sequence. The formula for the sum of an arithmetic sequence is given by:
Sn = (n/2)(2a1 + (n-1)d)
where:
Sn represents the sum of the first n terms,
a1 represents the first term,
n represents the number of terms, and
d represents the common difference.
Plugging in the values:
S10 = (10/2)(2(-22) + (10-1)(5))
S10 = 5(-44 + 9(5))
S10 = 5(-44 + 45)
S10 = 5(1)
S10 = 5
So, the sum of the first 10 terms of the arithmetic sequence is 5.