What is the Sum of the First 110 Terms in This Arithmetic Progression?

[demonelite123]

If the sum of the first 10 terms and the sum of the first 100 terms of a given arithmetic progression are 100 and 10, respectively, what is the sum of first 110 terms?

S(10) = (10/2) (a1 + a10) = 100
S(100) = (100/2) (a1 + a100) = 10

a1 + a10 = 20
a1 + a100 = 0.20

a100 = a1 + 99d
a10 = a1 + 9d

i subtracted and got:
a10 - a100 = 19.8
a1 + 9d - a1 - 99d = 19.8
d = -0.22

S(110) = (110/2) (a1 + a110)
a110 = a100 + 10d

S(110) = 55 (a1 + a100 - 2.2) = 55(0.20 - 2.2) = 55(-2) = -110

is this correct?

 

[Unco]

If the sum of the first 10 terms and the sum of the first 100 terms of a given arithmetic progression are 100 and 10, respectively, what is the sum of first 110 terms?

S(10) = (10/2) (a1 + a10) = 100
S(100) = (100/2) (a1 + a100) = 10

a1 + a10 = 20
a1 + a100 = 0.20

a100 = a1 + 99d
a10 = a1 + 9d

i subtracted and got:
a10 - a100 = 19.8
a1 + 9d - a1 - 99d = 19.8
d = -0.22

S(110) = (110/2) (a1 + a110)
a110 = a100 + 10d

S(110) = 55 (a1 + a100 - 2.2) = 55(0.20 - 2.2) = 55(-2) = -110

is this correct?

Yes; well done!

 

[Architeuthis Dux]

I would recommend double-checking your calculations and steps to ensure accuracy. However, your approach and reasoning seem to be correct. It is important to understand the given information and use it effectively to solve the problem. In this case, you have used the formula for the sum of an arithmetic progression and the given values to find the common difference (d) and then used it to find the sum of the first 110 terms. Overall, your solution seems to be reasonable.