Please help This is a proof assignment for my calc 2 class I

Please help! This is a proof assignment for my calc 2 class! I have no idea where to start.

A proof without words A picture is worth a thousand words, and sometimes the right picture can prove something must be true! Use the following picture to prove the formula for the sum of a geometric series with constant multiplier a > 0 and ratio 0

Solution

1) the sum of geometric series a + ar+ ar^2 + ... can be calculated using this picture.

the sum will the y-cordinate of intersecting point. as a is the length of each step corresponds to the each term in geometric series .

2) cordinate can be found by solving equation

y = a +rx , putting the value of x in it .

finally the last intersecting point can be found by solving

y = x and

y = a +rx

c)the length of each segment of zig-zag is the difference of y-cordinate or difference in x-cordinates .

for eg

the first vertical segment is of length a ,

y = a +rx , we put x = 0

y = a,

point on line y= x is (a,a)

similarly next vertical segment is of length ar

y = a +rx

y = a(1+r)

hence length = a(1+r) -a = ar

similarly others ar^2 ,ar^3 ...

4) nth partial sum = sum of first n vertical segment

5) sum of series = y-cordinate of intersection point of

y = a +rx & y =x

which is

y = a +ry

y(1-r) = a

or y = a/(1-r)

hence

a + ar+ ar^2 + ... = a/(1-r)