We are given the of a finite and want to select the choice that has the correct expression in .
14+20+26+32+38+44+50
To do it, we will find the sum of the given series and compare it with the sum of the series in each choice.
Since consecutive terms have a of 6, the given series is . To calculate its sum, we need to substitute the first and last terms a_1 = 14 and a_7 = 50, and the number of terms n= 7 in the formula for the .
We now know that the sum of the given series is 224. Therefore, the sum of the series of the correct choice must also be 224. Let's now calculate each sum one at a time.
.
∑^8_(n=2) (7n-1)
Since the formula is a , we know that the series is arithmetic. To calculate its sum, we need to find the first and last terms. Be aware that the series starts at n=2. This means that, to obtain the first term, we need to substitute n= 2 in the corresponding formula. Similarly, because the upper limit is 8, we obtain the last term by substituting n= 8.
| a_n=7n-1
|
| a_1=7( 2) - 1
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a_n=7( 8)-1
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| a_1=13
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a_n=55
|
Now we know that the first term is 13 and the last term 55. Conversely, the first and last terms of the given series are 14 and 50. Therefore, this series in summation notation does not represent the given series. Choice A is not the correct choice.
Choice B
We want to find the sum of the series in choice B.
∑^9_(n=3) (6n-4)
Since the formula is a linear function, we know that the series is arithmetic. To calculate its sum, we need to find the first and last terms. Be aware that the series starts at n=3. This means that, to obtain the first term, we need to substitute n= 3 in the corresponding formula. Similarly, because the upper limit is 9, we obtain the last term by substituting n= 9.
| a_n=6n-4
|
| a_1=6( 3)-4
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a_n=6( 9)-4
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| a_1=14
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a_n=50
|
Now we know that a_1=14 and a_n=50. Because the lower limit of the summation notation is 3 and the upper limit is 9, this series has 7 terms. We will calculate the sum of the series by substituting a_1= 14, a_n= 50, and n= 7 in the formula for the sum of a finite arithmetic series.
S_n=n/2(a_1+a_n)
S_7=7/2( 14+ 50)
S_7=3.5(14+50)
S_7=3.5(64)
S_7=224
Choice C
Consider the summation notation given in this part.
∑^8_(n=3) (6n-4)
Since the upper and lower limits are 8 and 3, respectively, this series has 6 terms. Conversely, the given series has 7 terms. Because they do not have the same number of terms, choice C does not represent the given series.
Choice D
We want to find the sum of the series in choice D.
∑^(14)_(n=8) (n+6)
Since the formula is a linear function, we know that the series is arithmetic. To calculate its sum, we need to find the first and last terms. Be aware that the series starts at n=8. This means that, to obtain the first term, we need to substitute n= 8 in the corresponding formula. Similarly, because the upper limit is 14, we obtain the last term by substituting n= 14.
| a_n=n+6
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| a_1= 8+6
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a_n= 14+6
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| a_1=14
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a_n=20
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Now we know that a_1=14 and a_n=20. The first term of the given series is 14, but the last term is 50. Therefore, this series in summation notation does not represent the given series. Choice D is not the correct choice.
