it is given that the sum of the seventh and tenth term of an arithmetic sequence is 12 and term 13 is equal to "-30"

Question

determine the sum of the first 100 terms in the sequence.

Answer 1

a+6d + a+9d = 12
a+12 = -30
solve for a and d, and then you want
S100 = 100/2 (2a+99d)

Answer 2

Since we know that the 13th term is -30, use the explicit formula for an arithmetic sequence, plugging in n=13 and solving for a1:
a(n) = a1+(n-1)d
a(13) = a1+(13-1)d
-30 = a1+12d
a1 = -30-12d

Since the 7th and 10th terms add up to 12, do the same thing:
a1+(7-1)d + a1+(10-1)d = 12
2(a1) + 6d + 9d = 12
2(a1) + 15d = 12
a1 + (15/2)d = 6
a1 = 6 - (15/2)d

Now, set the equations equal to each other to find d, the common difference:
-30 - 12d = 6 - (15/2)d
-30 = 6 + (9/2)d
-36 = (9/2)d
-8 = d

Lastly, find the value of a1:
a1 = 6 - (15/2)(-8)
a1 = 6 + 120/2
a1 = 6 + 60
a1 = 66

Therefore, your arithmetic sequence is a(n)=66+(n-1)(-8)

Using this information, we also know that the equation for an arithmetic series is S(n) = n[(a1 + a(n))/2], so we need to find what a(100) is and then we can determine S(100):

a(100) = 66 + (100-1)(-8)
a(100) = 66 + (99)(-8)
a(100) = 66 - 792
a(100) = -726

Therefore, we can determine S(100), which represents the sum of the first 100 terms in the sequence:
S(100) = 100[(66+a(100))/2]
S(100) = 100[(66-726)/2]
S(100) = 100[-660/2]
S(100) = 100(-330)
S(100) = -33000

In conclusion, the sum of the first 100 terms in the sequence is -33,000.

Answer 3

Hope this explanation helped!

Answer 4

thank you, for all the help