Question

An arithmetic series is represented by the equation (picture). Which of the following is true?

Answer 1

Answer:

C. The value of the series is equal to the value of the 1st or 24th numbers in the series.

Step-by-step explanation:

The given series is $S_{24}=\sum_{k=1}^{24}-6+0.5k$.

To find the first term of this series, we substitute k=1 to obtain:

$a_1=-6+0.5(1)=-6+0.5=-5.5$

To find the 24th term of the series, we put k=24 to get:

$a_{24}=-6+0.5(24)=-6+12=6$

The value of the series is the sum of all the terms in the series.

We can find the value of the series using the formula:

$s_{n}=\frac{n}{2}(a_1+l)$.

In this case, the last term is $a_{24}=l$.

$\implies s_{24}=\frac{24}{2}(-5.5+6)$.

$\implies s_{24}=12(0.5)$.

$\implies s_{24}=6$.

Now let us analyse the options:

A. The value of the series is greater than the value of the 24th number in the series.

False: because 6 is not greater than 6.

B. The value of the series is between the values of the 1st and 24th numbers in the series.

False:  because 6 is not between -5.5 and 6

C. The value of the series is equal to the value of the 1st or 24th numbers in the series.

True: In logics, an "or" statement (disjunction) is true if one of the alternatives is true. In this case the sum of the series is equal to the 24th number of the series because 6=6 is true

D. The value of the series is less than the value of the 1st number in the series.

False: because 6 is not less than -5.5

Answer 2

The correct answer is C; the value of the series shown in the picture is equal to the value of the 1st or 24th numbers in the series.

Further Explanation

Each value in the series can be determined by substituting the value of k into the expression.  To begin, substitute 1 in place of k for the first term:

• -6+0.5(1) = -6+0.5 = -5.5

Continuing with k = 2,

• -6+0.5(2) = -6+1 = -5
• -6+0.5(3) = -6+1.5 = -4.5
• -6+0.5(4) = -6+2 = -4
• -6+0.5(5) = -6+2.5 = -3.5
• -6+0.5(6) = -6+3 = -3
• -6+0.5(7) = -6+3.5 = -2.5
• -6+0.5(8) = -6+4 = -2
• -6+0.5(9) = -6+4.5 = -1.5
• -6+0.5(10) = -6+5 = -1
• -6+0.5(11) = -6+5.5 = -0.5
• -6+0.5(12) = -6+6 = 0
• -6+0.5(13) = -6+6.5 = 0.5
• -6+0.5(14) = -6+7 = 1
• -6+0.5(15) = -6+7.5 = 1.5
• -6+0.5(16) = -6+8 = 2
• -6+0.5(17) = -6+8.5 = 2.5
• -6+0.5(18) = -6+9 = 3
• -6+0.5(19) = -6+9.5 = 3.5
• -6+0.5(20) = -6+10 = 4
• -6+0.5(21) = -6+10.5 = 4.5
• -6+0.5(22) = -6+11 = 5
• -6+0.5(23) = -6+11.5 = 5.5
• -6+0.5(24) = -6+12 = 6

The value of the series is given by adding together the terms of the series.  Doing this, we see we have matching positive and negative values; these will sum to 0.  Taking all of these matching values out, we are left with the number 6.  This is the value of the series.  This is the value of the 24th number in the series; this means the correct answer is C.

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Keywords:  arithmetic series, value of series, value of arithmetic series