Use Gauss's approach to find the following sums (do not use formulas).
a. 1 + 2 + 3 + 4 +...+ 102
b. 1 + 3 + 5 + 7 +...+1001
a. The sum of the sequence is ?
My answer 102-51(103)= 5,253
b. The sum of the sequence is?
My answer 500.3(1,002)=501,300.6
Are my answers correct? If not why, and how do I get the correct answers?
Yes, your answers are correct. Gauss's approach involves adding the first and last numbers in the sequence and then multiplying the result by the number of terms in the sequence. For the first sequence, the first number is 1 and the last number is 102, so the sum is (1 + 102) x (102/2) = 5,253. For the second sequence, the first number is 1 and the last number is 1001, so the sum is (1 + 1001) x (501/2) = 501,300.6.
Your answer for part a is not correct. To find the sum of the sequence 1 + 2 + 3 + 4 + ... + 102, you can use Gauss's approach which involves finding the average of the first and the last term, and then multiplying it by the number of terms. The formula for the sum of an arithmetic sequence is: Sn = (n/2)(a1 + an), where Sn is the sum, n is the number of terms, a1 is the first term, and an is the last term.
In this case, the first term, a1, is 1 and the last term, an, is 102. The number of terms, n, can be found by subtracting the first term from the last term and adding 1: n = an - a1 + 1 = 102 - 1 + 1 = 102. Plugging these values into the formula, we get:
Sn = (102/2)(1 + 102)
= 51(103)
= 5,253
So the correct sum for the sequence 1 + 2 + 3 + 4 + ... + 102 is 5,253.
For part b, the sequence 1 + 3 + 5 + 7 + ... + 1001 is an arithmetic sequence with a common difference of 2. To find the sum using Gauss's approach, we need to find the number of terms in the sequence. Since 1 is the first term and 1001 is the last term, we can find the number of terms, n, by solving the equation:
an = a1 + (n-1)d
1001 = 1 + (n-1)2
1000 = 2(n-1)
500 = n-1
n = 501
So there are 501 terms in the sequence. Now we can find the sum using the formula Sn = (n/2)(a1 + an):
Sn = (501/2)(1 + 1001)
= 250.5(1,002)
= 251,251
Therefore, the correct sum for the sequence 1 + 3 + 5 + 7 + ... + 1001 is 251,251.
Your answers are not correct. Let's go step-by-step to find the correct answers using Gauss's approach.
a. To find the sum of the sequence 1 + 2 + 3 + 4 +...+ 102, follow these steps:
Step 1: Find the number of terms in the sequence by subtracting the first term (1) from the last term (102) and adding 1: 102 - 1 + 1 = 102.
Step 2: Divide the number of terms by 2: 102 / 2 = 51.
Step 3: Multiply the result from Step 2 by the sum of the first and last term: 51 * (1 + 102) = 5,253.
So, the correct answer is 5,253.
b. To find the sum of the sequence 1 + 3 + 5 + 7 +...+ 1001, follow these steps:
Step 1: Find the number of terms in the sequence. Since each term increases by 2, we can find the number of terms by dividing the difference between the last term (1001) and the first term (1) by 2 and adding 1: (1001 - 1) / 2 + 1 = 501.
Step 2: Multiply the number of terms by the sum of the first and last term: 501 * (1 + 1001) = 501 * 1002 = 501,502.
So, the correct answer is 501,502.
Therefore, the correct answers are:
a. The sum of the sequence 1 + 2 + 3 + 4 +...+ 102 is 5,253.
b. The sum of the sequence 1 + 3 + 5 + 7 +...+ 1001 is 501,502.