Let s(n)=1+2+···+ n be the sum of the first n natural numbers and let c(n)=1 3 +2 3 + ···+ n3 be the sum of the first n cubes. prove the following equalitiesby inductionon n, to arriveat thecurious conclusionthatc(n)=s2(n) for every

Step-by-step explanation:

We have

To prove that by induction

Let P(n) be that statement for n

n=1 gives left side =1 =right side

True for n=1

Assume that P(k) is true

i.e.

We have to prove P(k+1) is true assuming P(k)

LHS of

RHS =

since by adding last term with Ck gives the sum

For simplification take (k+1) square outside

RHS =

Thus proved by mathematical induction

step-by-step explanation:

i found the life tables for the year 2012. 

life table of males at age 30. probability of dying between ages x and x+1

total population:  

29–30 . . . . . . . . . . . . 0.001030

30–31 . . . . . . . . . . . . 0.001061

males:  

29–30 . . . . . . . . . . . . 0.001420

30–31 . . . . . . . . . . . . 0.001448

white population

29–30 . . . . . . . . . . . . 0.001005

30–31 . . . . . . . . . . . . 0.001037

white males

29–30 . . . . . . . . . . . . 0.001372

30–31 . . . . . . . . . . . . 0.001399

non hispanic white population

29–30 . . . . . . . . . . . . 0.001083

30–31 . . . . . . . . . . . . 0.001116

non hispanic white males

29–30 . . . . . . . . . . . . 0.001477

30–31 . . . . . . . . . . . . 0.001503

jacob, non hispanic white male. 30 years old.

probability that jacob lives to be 30 years old. 

since the given probabilities are probabilities of dying, i think we simply subtract it from 1 to get the probability of living. we will use the probability of non hispanic white male.

                                          probability of living to be 30 y.o.

29 - 30 : 1 - 0.001477 = 0.998523

30 - 31 : 1 - 0.001503 = 0.998497