QuestionFind the sums of these sequences: a. $1+2+3+\ldots+1002$ b. $1+3+5+\ldots+999$ c. $6+13+20+\ldots+706$ d. $494+489+484+\ldots+4$

Studdy Solution

STEP 1

Assumptions1. We are dealing with arithmetic sequences. . We are using Gauss's approach to find the sum of the sequences.
3. Gauss's approach involves pairing terms from the start and end of the sequence that add up to the same total.

STEP 2

We start with the sequence $1+2++4+\ldots+1002$ . The first and last terms of the sequence are1 and1002 respectively.

STEP 3

According to Gauss's approach, we pair the first and last terms, the second and second last terms, and so on. Each pair sums up to the same total.
$1 +1002 =1003$ $2 +1001 =1003$ $3 +1000 =1003$ $\vdots$

STEP 4

The sequence has1002 terms. Since we are pairing the terms, we have501 pairs.

STEP 5

The sum of the sequence is the sum of all pairs, which is the sum of one pair times the number of pairs.
$Sum = Number\, of\, pairs \\times Sum\, of\, one\, pair$

STEP 6

Plug in the values for the number of pairs and the sum of one pair to calculate the sum of the sequence.
$Sum =501 \\times1003$

STEP 7

Calculate the sum of the sequence.
$Sum =501 \\times1003 =502503$ The sum of the sequence $1+2+3+4+\ldots+1002$ is502503.

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QuestionFind the sums of these sequences: a. $1+2+3+\ldots+1002$ b. $1+3+5+\ldots+999$ c. $6+13+20+\ldots+706$ d. $494+489+484+\ldots+4$

Studdy Solution

STEP 1

Assumptions1. We are dealing with arithmetic sequences. . We are using Gauss's approach to find the sum of the sequences.
3. Gauss's approach involves pairing terms from the start and end of the sequence that add up to the same total.

STEP 2

We start with the sequence $1+2++4+\ldots+1002$ . The first and last terms of the sequence are1 and1002 respectively.

STEP 3

According to Gauss's approach, we pair the first and last terms, the second and second last terms, and so on. Each pair sums up to the same total.
$1 +1002 =1003$ $2 +1001 =1003$ $3 +1000 =1003$ $\vdots$

STEP 4

The sequence has1002 terms. Since we are pairing the terms, we have501 pairs.

STEP 5

The sum of the sequence is the sum of all pairs, which is the sum of one pair times the number of pairs.
$Sum = Number\, of\, pairs \\times Sum\, of\, one\, pair$

STEP 6

Plug in the values for the number of pairs and the sum of one pair to calculate the sum of the sequence.
$Sum =501 \\times1003$

STEP 7

Calculate the sum of the sequence.
$Sum =501 \\times1003 =502503$ The sum of the sequence $1+2+3+4+\ldots+1002$ is502503.

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QuestionFind the sums of these sequences: a. 1+2+3+…+10021+2+3+\ldots+10021+2+3+…+1002 b. 1+3+5+…+9991+3+5+\ldots+9991+3+5+…+999 c. 6+13+20+…+7066+13+20+\ldots+7066+13+20+…+706 d. 494+489+484+…+4494+489+484+\ldots+4494+489+484+…+4

Studdy Solution

STEP 1

Assumptions1. We are dealing with arithmetic sequences. . We are using Gauss's approach to find the sum of the sequences.3. Gauss's approach involves pairing terms from the start and end of the sequence that add up to the same total.

STEP 2

We start with the sequence 1+2++4+…+10021+2++4+\ldots+10021+2++4+…+1002. The first and last terms of the sequence are1 and1002 respectively.

STEP 3

According to Gauss's approach, we pair the first and last terms, the second and second last terms, and so on. Each pair sums up to the same total.1+1002=10031 +1002 =10031+1002=10032+1001=10032 +1001 =10032+1001=10033+1000=10033 +1000 =10033+1000=1003⋮\vdots⋮

STEP 4

The sequence has1002 terms. Since we are pairing the terms, we have501 pairs.

STEP 5

The sum of the sequence is the sum of all pairs, which is the sum of one pair times the number of pairs.Sum=Number of pairstimesSum of one pairSum = Number\, of\, pairs \\times Sum\, of\, one\, pairSum=NumberofpairstimesSumofonepair

STEP 6

Plug in the values for the number of pairs and the sum of one pair to calculate the sum of the sequence.Sum=501times1003Sum =501 \\times1003Sum=501times1003

STEP 7

Calculate the sum of the sequence.Sum=501times1003=502503Sum =501 \\times1003 =502503Sum=501times1003=502503The sum of the sequence 1+2+3+4+…+10021+2+3+4+\ldots+10021+2+3+4+…+1002 is502503.

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionFind the sums of these sequences: a. $1+2+3+\ldots+1002$ b. $1+3+5+\ldots+999$ c. $6+13+20+\ldots+706$ d. $494+489+484+\ldots+4$

Assumptions1. We are dealing with arithmetic sequences. . We are using Gauss's approach to find the sum of the sequences.
3. Gauss's approach involves pairing terms from the start and end of the sequence that add up to the same total.

We start with the sequence $1+2++4+\ldots+1002$ . The first and last terms of the sequence are1 and1002 respectively.

According to Gauss's approach, we pair the first and last terms, the second and second last terms, and so on. Each pair sums up to the same total.
$1 +1002 =1003$ $2 +1001 =1003$ $3 +1000 =1003$ $\vdots$

The sum of the sequence is the sum of all pairs, which is the sum of one pair times the number of pairs.
$Sum = Number\, of\, pairs \\times Sum\, of\, one\, pair$

Plug in the values for the number of pairs and the sum of one pair to calculate the sum of the sequence.
$Sum =501 \\times1003$

Calculate the sum of the sequence.
$Sum =501 \\times1003 =502503$ The sum of the sequence $1+2+3+4+\ldots+1002$ is502503.