QuestionCompare the sums $O = 3 + 5 + 7 + \ldots + 103$ and $E = 4 + 6 + 8 + \ldots + 104$ . Which is greater and by how much?

Studdy Solution

STEP 1

Assumptions1. The series O represents the sum of odd numbers from3 to103. . The series represents the sum of even numbers from4 to104.
3. Both series are arithmetic series, with a common difference of.

STEP 2

First, we need to find the number of terms in each series. The formula to find the number of terms in an arithmetic series is $n = \frac{Last\, term - First\, term}{Common\, difference} +1$

STEP 3

Calculate the number of terms in series O.
$n = \frac{103 -3}{2} +1$

STEP 4

Calculate the number of terms in series.
$n_E = \frac{104 -4}{2} +1$

STEP 5

The sum of an arithmetic series can be found using the formula $Sum = \frac{n}{2} \times (First\, term + Last\, term)$

STEP 6

Calculate the sum of series O without actually computing the sum.
$= \frac{n}{2} \times (3 +103)$

STEP 7

Calculate the sum of series without actually computing the sum.
$= \frac{n_E}{2} \times (4 +104)$

STEP 8

Now, we need to find which series has a greater sum and by how much. We can do this by subtracting the smaller sum from the larger sum.
If $>$ , then $Difference = O -$
If $> O$ , then $Difference = - O$

STEP 9

Finally, simplify the difference to find which series has a greater sum and by how much.

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QuestionCompare the sums $O = 3 + 5 + 7 + \ldots + 103$ and $E = 4 + 6 + 8 + \ldots + 104$ . Which is greater and by how much?

Studdy Solution

STEP 1

Assumptions1. The series O represents the sum of odd numbers from3 to103. . The series represents the sum of even numbers from4 to104.
3. Both series are arithmetic series, with a common difference of.

STEP 2

First, we need to find the number of terms in each series. The formula to find the number of terms in an arithmetic series is $n = \frac{Last\, term - First\, term}{Common\, difference} +1$

STEP 3

Calculate the number of terms in series O.
$n = \frac{103 -3}{2} +1$

STEP 4

Calculate the number of terms in series.
$n_E = \frac{104 -4}{2} +1$

STEP 5

The sum of an arithmetic series can be found using the formula $Sum = \frac{n}{2} \times (First\, term + Last\, term)$

STEP 6

Calculate the sum of series O without actually computing the sum.
$= \frac{n}{2} \times (3 +103)$

STEP 7

Calculate the sum of series without actually computing the sum.
$= \frac{n_E}{2} \times (4 +104)$

STEP 8

Now, we need to find which series has a greater sum and by how much. We can do this by subtracting the smaller sum from the larger sum.
If $>$ , then $Difference = O -$
If $> O$ , then $Difference = - O$

STEP 9

Finally, simplify the difference to find which series has a greater sum and by how much.

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QuestionCompare the sums O=3+5+7+…+103O = 3 + 5 + 7 + \ldots + 103O=3+5+7+…+103 and E=4+6+8+…+104E = 4 + 6 + 8 + \ldots + 104E=4+6+8+…+104. Which is greater and by how much?

Studdy Solution

STEP 1

Assumptions1. The series O represents the sum of odd numbers from3 to103. . The series represents the sum of even numbers from4 to104.3. Both series are arithmetic series, with a common difference of.

STEP 2

First, we need to find the number of terms in each series. The formula to find the number of terms in an arithmetic series isn=Last term−First termCommon difference+1n = \frac{Last\, term - First\, term}{Common\, difference} +1n=CommondifferenceLastterm−Firstterm​+1

STEP 3

Calculate the number of terms in series O.n=103−32+1n = \frac{103 -3}{2} +1n=2103−3​+1

STEP 4

Calculate the number of terms in series.nE=104−42+1n_E = \frac{104 -4}{2} +1nE​=2104−4​+1

STEP 5

The sum of an arithmetic series can be found using the formulaSum=n2×(First term+Last term)Sum = \frac{n}{2} \times (First\, term + Last\, term)Sum=2n​×(Firstterm+Lastterm)

STEP 6

Calculate the sum of series O without actually computing the sum.=n2×(3+103) = \frac{n}{2} \times (3 +103)=2n​×(3+103)

STEP 7

Calculate the sum of series without actually computing the sum.=nE2×(4+104) = \frac{n_E}{2} \times (4 +104)=2nE​​×(4+104)

STEP 8

Now, we need to find which series has a greater sum and by how much. We can do this by subtracting the smaller sum from the larger sum.If > >>, then Difference=O−Difference = O -Difference=O−If >O > O>O, then Difference=−ODifference = - ODifference=−O

STEP 9

Finally, simplify the difference to find which series has a greater sum and by how much.

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Studdy solves anything!

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

QuestionCompare the sums $O = 3 + 5 + 7 + \ldots + 103$ and $E = 4 + 6 + 8 + \ldots + 104$ . Which is greater and by how much?

Assumptions1. The series O represents the sum of odd numbers from3 to103. . The series represents the sum of even numbers from4 to104.
3. Both series are arithmetic series, with a common difference of.

First, we need to find the number of terms in each series. The formula to find the number of terms in an arithmetic series is $n = \frac{Last\, term - First\, term}{Common\, difference} +1$

Calculate the number of terms in series O.
$n = \frac{103 -3}{2} +1$

Calculate the number of terms in series.
$n_E = \frac{104 -4}{2} +1$

The sum of an arithmetic series can be found using the formula $Sum = \frac{n}{2} \times (First\, term + Last\, term)$

Calculate the sum of series O without actually computing the sum.
$= \frac{n}{2} \times (3 +103)$

Calculate the sum of series without actually computing the sum.
$= \frac{n_E}{2} \times (4 +104)$

Now, we need to find which series has a greater sum and by how much. We can do this by subtracting the smaller sum from the larger sum.
If $>$ , then $Difference = O -$
If $> O$ , then $Difference = - O$