Find the value of 100-98+96-94+92-90+…+8-6+4-2
The Correct Answer and Explanation is:
Let’s break down the sequence:
The given expression is:
[
100 – 98 + 96 – 94 + 92 – 90 + \dots + 8 – 6 + 4 – 2
]
This is a sequence of alternating subtraction and addition of consecutive even numbers. We can pair the terms as follows:
[
(100 – 98) + (96 – 94) + (92 – 90) + \dots + (4 – 2)
]
Now, simplify each pair:
[
100 – 98 = 2, \quad 96 – 94 = 2, \quad 92 – 90 = 2, \quad \dots, \quad 4 – 2 = 2
]
Each pair results in a sum of 2. Let’s now count how many pairs there are. The sequence starts at 100 and ends at 2, and each term in the sequence is an even number. So, we have the even numbers:
[
100, 98, 96, 94, \dots, 4, 2
]
These numbers form an arithmetic sequence with a first term of 100, a common difference of -2, and a last term of 2. To find the number of terms in this sequence, we use the formula for the nth term of an arithmetic sequence:
[
a_n = a_1 + (n – 1) \cdot d
]
Where:
- (a_n) is the nth term,
- (a_1) is the first term,
- (d) is the common difference,
- (n) is the number of terms.
Substitute the values:
[
2 = 100 + (n – 1) \cdot (-2)
]
Simplifying:
[
2 = 100 – 2n + 2
]
[
2n = 100
]
[
n = 50
]
So, there are 50 terms in this sequence. Since we are pairing each consecutive term, we have 25 pairs. Each pair sums to 2, so the total sum is:
[
25 \times 2 = 50
]
Thus, the value of the given expression is:
[
\boxed{50}
]
Explanation:
We paired consecutive terms from the sequence and simplified them to obtain pairs of 2. The sequence of even numbers from 100 to 2 contains 50 terms, which gives us 25 pairs, each summing to 2. Therefore, the total sum of the sequence is 50.