Use Gauss's approach to find the following sums ( do not use formulas). A. 1+2+3+4+...1001
B. 1+3+5+7...+99.
To find the sum using Gauss's approach, we can apply the formula for the sum of an arithmetic series.
A. The sum of the series 1+2+3+4+...+1001 can be found using the formula Sn = (n/2)(a + L), where Sn represents the sum of the series, n is the number of terms, a is the first term, and L is the last term.
In this case, the first term a = 1, and the last term L = 1001. We need to find the value of n, which represents the number of terms in the series.
We can use the formula for the nth term of an arithmetic sequence, an = a + (n - 1)d, where an is the nth term, and d is the common difference.
In this series, the common difference is 1 (since we are adding 1 to each term).
Using the formula, we can find the value of n:
1001 = 1 + (n - 1)1
1001 = 1 + n - 1
1001 = n
Now, we have the value of n = 1001. Plugging this into the formula for the sum of the arithmetic series, we get:
Sn = (n/2)(a + L)
Sn = (1001/2)(1 + 1001)
Sn = (1001/2)(1002)
Calculating this gives us:
Sn = 501 * 1002 = 501,502
Therefore, the sum of the series 1+2+3+4+...+1001 is 501,502.
B. The sum of the series 1+3+5+7+...+99 can also be found using Gauss's approach. However, this series does not have a constant common difference, as we are adding 2 to each term.
Instead of trying to find the number of terms (n) and then using the formula for the sum of an arithmetic series, we can directly find the sum by dividing the series into pairs with equal sums.
The first pair is 1 + 99 = 100
The second pair is 3 + 97 = 100
The third pair is 5 + 95 = 100
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The last pair is 99, which does not have another term to pair with.
Since we divided the series into pairs, we have a total of (99 - 1) / 2 + 1 pairs. That gives us (98 / 2) + 1 = 49 + 1 = 50 pairs.
Now, we need to find the sum of each pair, which is 100. Multiplying the sum of each pair (100) by the number of pairs (50), we get:
Sum = 100 * 50 = 5000
Therefore, the sum of the series 1+3+5+7+...+99 is 5000.
To find the sum of the consecutive integers from 1 to 1001, we can use Gauss's approach, which involves pairing the first and last terms, the second and second-to-last terms, and so on. This approach allows us to find the sum more efficiently.
A. Sum of 1+2+3+4+...+1001:
Step 1: Determine the total number of terms.
Since we are summing consecutive integers, the total number of terms is equal to the last term (1001) minus the first term (1), plus 1:
Number of terms = 1001 - 1 + 1 = 1001.
Step 2: Pair the terms.
Pair the first term (1) with the last term (1001), the second term (2) with the second-to-last term (1000), and so on.
Step 3: Calculate the sum of each pair.
The sum of each pair is always equal to 1002 (1 + 1001 = 2 + 1000 = 3 + 999 = ... = 1002).
Step 4: Determine the number of pairs.
Since we are pairing the terms, the number of pairs is half of the total number of terms:
Number of pairs = Number of terms / 2 = 1001 / 2 = 500.5.
Note: We have a half-pair (the middle term) when the total number of terms is odd.
Step 5: Calculate the sum.
The sum can be found by multiplying the sum of each pair by the number of pairs:
Sum = Sum of each pair × Number of pairs = 1002 × 500.5 = 501,501.
Therefore, the sum of the consecutive integers 1+2+3+4+...+1001 is 501,501.
B. Sum of 1+3+5+7+...+99:
Step 1: Determine the total number of terms.
Since the numbers are odd and increasing by 2 each time, the last term is 99. The first term is 1. So we can calculate the number of terms using the formula:
Number of terms = (Last term - First term)/Difference + 1 = (99 - 1)/2 + 1 = 49 + 1 = 50.
Step 2: Pair the terms.
Pair the first term (1) with the last term (99), the second term (3) with the second-to-last term (97), and so on.
Step 3: Calculate the sum of each pair.
The sum of each pair is always equal to 100 (1 + 99 = 3 + 97 = 5 + 95 = ... = 100).
Step 4: Determine the number of pairs.
Since we are pairing the terms, the number of pairs is half of the total number of terms:
Number of pairs = Number of terms / 2 = 50 / 2 = 25.
Step 5: Calculate the sum.
The sum can be found by multiplying the sum of each pair by the number of pairs:
Sum = Sum of each pair × Number of pairs = 100 × 25 = 2500.
Therefore, the sum of the odd integers from 1+3+5+7+...+99 is 2500.
Just match the values in pairs.
A. Each pair adds up to 1002
B. Each pair adds up to 100
Now just figure out how many pairs there are.