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

In summary, an arithmetic sequence is a sequence of numbers with a constant difference between consecutive terms. To find the common difference, subtract any two consecutive terms. The formula for finding the nth term is a<sub>n</sub> = a<sub>1</sub> + (n-1)d, where a<sub>n</sub> is the nth term, a<sub>1</sub> is the first term, and d is the common difference. To solve a word problem, identify the given information and use the formula for finding the missing term(s). The sum of an arithmetic sequence is calculated using the formula S<sub>n</sub> = n/2(a<sub>1</sub> + a<
  • #1
demonelite123
219
0
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?
 
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  • #2
demonelite123 said:
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!
 
  • #3


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.
 

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

1. What is an arithmetic sequence?

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is known as the common difference.

2. How do I find the common difference of an arithmetic sequence?

To find the common difference, subtract any two consecutive terms in the sequence. The result will be the common difference.

3. What is the formula for finding the nth term of an arithmetic sequence?

The formula for finding the nth term of an arithmetic sequence is:
an = a1 + (n-1)d
where an is the nth term, a1 is the first term, and d is the common difference.

4. How do I solve a word problem involving arithmetic sequences?

To solve a word problem involving arithmetic sequences, first identify the given information such as the first term, common difference, and number of terms. Then, use the formula for finding the nth term to find the missing term or terms.

5. What is the sum of an arithmetic sequence?

The sum of an arithmetic sequence is calculated by using the formula:
Sn = n/2(a1 + an)
where Sn is the sum of the first n terms, a1 is the first term, and an is the nth term.

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