QuestionFind the sums of these sequences using Gauss's method (no formulas): a. b. c. d.
Studdy Solution
STEP 1
Assumptions1. We are using Gauss's approach to find the sums of the arithmetic sequences.
. We are not using any formulas to solve these sequences.
3. The sequences are as follows a.
b.
c.
d.
STEP 2
Gauss's approach to summing an arithmetic sequence involves pairing terms from the start and end of the sequence that add up to the same value.For sequence a, we pair the first and last terms, the second and second-to-last terms, and so on. Each pair sums to .
STEP 3
We can find the number of pairs by dividing the total number of terms by . The total number of terms in sequence a is .
STEP 4
Since the number of terms is odd, we have one term (the middle term) that is not part of a pair. This term is .
STEP 5
The sum of the sequence is the sum of all the pairs plus the middle term.
STEP 6
Substitute the values into the formula to find the sum of sequence a.
STEP 7
Calculate the sum of sequence a.
STEP 8
For sequence b, we pair the first and last terms, the second and second-to-last terms, and so on. Each pair sums to .
STEP 9
We can find the number of pairs by dividing the total number of terms by . The total number of terms in sequence b is .
STEP 10
Since the number of terms is odd, we have one term (the middle term) that is not part of a pair. This term is .
STEP 11
Substitute the values into the formula to find the sum of sequence b.
STEP 12
Calculate the sum of sequence b.
STEP 13
For sequence c, we pair the first and last terms, the second and second-to-last terms, and so on. Each pair sums to .
STEP 14
We can find the number of pairs by dividing the total number of terms by . The total number of terms in sequence c is .
STEP 15
Since the number of terms is even, there are no unpaired terms.
STEP 16
Substitute the values into the formula to find the sum of sequence c.
STEP 17
Calculate the sum of sequence c.
STEP 18
For sequence d, we pair the first and last terms, the second and second-to-last terms, and so on. Each pair sums to .
STEP 19
We can find the number of pairs by dividing the total number of terms by $$. The total number of terms in sequence d is $100$.
STEP 20
Since the number of terms is even, there are no unpaired terms.
STEP 21
Substitute the values into the formula to find the sum of sequence d.
STEP 22
Calculate the sum of sequence d.
The sums of the sequences are as followsa.
b.
c.
d.
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