Question 1.11: (a) The sum of the first n terms of an arithmetic series is ......
(a) The sum of the first n terms of an arithmetic series is calculated by the formula
S_{n}={\frac{n}{2}}\left(2a+(n-1)d\right)
Calculate the value of S_n when n = 35, a = 200 and d=−2.5.
Note: The subscript in S_n simply indicates that S is the sum of n terms: the sum of 35 terms is written symbolically as S_{35}.
(b) In statistics, the formula for the intercept, a of a least-squares line is given by the formula
a=\frac{\sum y – b\sum x}{n}
Evaluate a when n = 9, \sum y = 3, b = −2.35 and \sum x = -21
(a) Substitute the values given into the formula
S_n = \frac{n}{2}(2a + (n – 1)d) evaluate the part of the formula within the bracket first
S_{35} = \frac{35}{2}(2 × 200 + (35 – 1) × (-2.5))use a bracket when substituting the negative value, −2.5
S_{35} = \frac{35}{2}(400 + (34) × (-2.5))S_{35} = \frac{35}{2}(400 – (34) × (2.5)) multiplying two unlike signs results in a negative value
S_{35} = \frac{35}{2}(400 – (85))S_{35} = \frac{35}{2}(315) subtract the two numbers within the bracket
S_{35} = 17.5 (315) = 5512.5 multiplying by the 35/2 outside the bracket
(b) Substitute n = 9, \sum y = 3, b = −2.35 and \sum x = -21 into the formula
a = \frac{\sum y – b \sum x}{n} using brackets when substituting negative quantities
a = \frac{3 – ( -2.35)(-21)}{9} evaluate the top line (numerator) to a single figure
a = \frac{3 – ((49.35)}{9} multiplying two like signs gives a positive number
a = -\frac{(46.35)}{9} divide the numerator by the denominator
a = −5.15