-90+30+-10+(10/3)... Find the sum of the first 10 terms ( Express as a fraction)
Have you figured out the pattern?
To find the sum of the first 10 terms of the given sequence, we need to add up all the terms from the 1st term to the 10th term. Let's break down the problem step by step:
First, let's write down the given sequence:
-90 + 30 + (-10) + (10/3) + ...
Now, let's find out what the pattern is for the sequence. We can observe that the pattern is as follows:
1st term: -90
2nd term: 30
3rd term: -10
4th term: 10/3
The sequence alternates between -90, 30, -10, 10/3. We can notice that the sign alternates between positive and negative, and we also have a constant term "10".
Now, we can write the terms explicitly:
1st term: -90
2nd term: 30
3rd term: -10
4th term: 10/3
5th term: -10
6th term: 10/3
7th term: -10
8th term: 10/3
9th term: -10
10th term: 10/3
To find the sum of these terms, we will add them up. Let's denote the sum of the first 10 terms as S10. We can calculate it as follows:
S10 = -90 + 30 + (-10) + (10/3) + (-10) + (10/3) + (-10) + (10/3) + (-10) + (10/3)
Simplifying the expression, we can combine like terms:
S10 = -90 + (-10 - 10 - 10 - 10 - 10) + (30 + (10/3) + (10/3) + (10/3) + (10/3))
S10 = -90 + (-50) + (30 + 4*(10/3))
S10 = -90 - 50 + (30 + (40/3))
S10 = -140 + (90 + 40/3)
S10 = (-140 + 90) + 40/3
S10 = -50 + (40/3)
To express this sum as a fraction, we need a common denominator. The common denominator of 1 and 3 is 3. So, we can rewrite the expression:
S10 = -50 + (40/3)
S10 = -50 + (40/3)*(3/3)
S10 = -50 + 120/3
S10 = -50 + 40
S10 = -10
Therefore, the sum of the first 10 terms of the given sequence is -10, expressed as a fraction.