If the first and the last terms of an arithmetic series are 10 and 62, show that the sum of the series varies directly as the number of terms.

This shows that the sum of the series varies directly as the number of terms.

Step-by-step explanation:

The terms of the airthmetic series is given by .

Where is the first term .

n represented the number of terms in the airthmetic series .

d is the common difference .

As given

If the first and the last terms of an arithmetic series are 10 and 62 .

Putting in the above

62 - 10 = (n-1)d

52 = (n-1)d

The Sum of the nth terms of the airthmetic series is given by .

Putting the values in the above

(This is for the n terms )

Therefore this shows that the sum of the series varies directly as the number of terms.

let soccer = x and golf = y

part a

1st equation: x + y = 125

2nd equation: x = y +45

part b

replace x in the 1st equation with the 2nd one:

y + 45 + y = 125

simplify:

2y + 45 = 125

subtract 45 from each side:

2y = 80

divide both sides by 2:

y = 80/2

y = 40

she plays golf for 40 minutes.

part c

no because she plays soccer for 45 minutes longer than golf 40 +45 =   85 minutes, not 80 minutes.