Progressions

AM = (a1+a2+a3+......+an)/n
=(n(a1+an)/2)/n
=(a1+an)/2 = (2 + 30)/2 = 16

Sum of AP = (n/2)[2a+(n-1)d]
= 4*[4+7*4]
=128

AM = (a1+a2+a3+......+an)/n =(n(a1+an)/2)/n =a1+a2/2 =10-2/2 = 4

Sum of series = (n/2)[2a+(n-1)d]
= 20

sum of AP = (n/2)[2a+(n-1)d]
n=19, a=2, d=1/2
S = (19/2)[2*2+(19-1)1/2]
=(19/2)[4+9]
=9.5*13 = 123.5

AM of Series: (2+11)/2
=13/2

Sol: Sn=[a(1-rⁿ)]/(1-r) =[1(1-3¹⁰)]/(1-3) =[(1-3¹⁰)]/(1-3)

Sn=a/(1-r)
= (1/3)/(1-1/3)
=1/2

Clearly, since the given integers are positive, their sum can't be negative. Also, since the numbers are all integers their sum can't be a fraction. Let's take 1, 3 and 9. The product of these three integers is 27 = 3³. This can also be written as nⁿ where n=3. As we can see, the sum of these 3 integers is not equal to 3. Therefore, we are left with the fourth option.

The total distance traveled by the ball is the sum of two infinite series: a. Series 1: the distance traveled by the ball when it's falling down b. Series 2: the distance traveled by the ball when it's bouncing up S1 = a1 / (1 - r1) and S2 = a2 / (1 - r2) S1 = 180 / (1 - 3/5) and S2 = (180 * 3/5) / (1 - 3/5) S1 = 180 / (2/5) and S2 = 108 / (2/5) S1 = 180 * 5/2 and S2 = 108 * 5/2 S1 = 450 and S2 = 270 Therefore, S = S1+S2 = 720 m.