Chapter 4.1, Problem 75SR

(a)

To determine

To Calculate:

Value of the 10^th question , if the value of the 1^st question is $3000.

Answer to Problem 75SR

The value of the 10^th question is worth $25,500.

Explanation of Solution

Given Information :

Value of 1^st question = $3000 = first term of AP = a

Formula Used:

For A.P

  $a,a+d,a+2d,\mathrm{........}$

n-th term is given by:

  $\begin{array}{l}{a}_n=a+(n-1)d\mathrm{...........................}(1)\\ where\\ a=first\_term\_of\_AP\\ d=common\_difference\end{array}$

Sum of n terms is given by:

  $\begin{array}{l}{S}_n=\frac{n}{2}[2a+(n-1)d]\\ \Rightarrow S_n=\frac{n}{2}[a+a_n]\mathrm{........................}(2)\end{array}$

Calculation:

Here, we have the Arithematic Progression (AP) as follows:

q = Number of questions

a = value of 1^st question = 3000

d = the common difference = 2500( Since each successive question’s value increases by $2500)

Hence, the AP is as follows :

3000 , 3000+2500, 3000+2(2500) , ………

So, we find the 10^th term of about Arithematic Progression :

  $\begin{array}{l}{a}_q=a+(q-1)d\mathrm{...............................}[qth-term]\\ \Rightarrow a_q=3000+(q-1)\mathrm{2500...............}[a=3000,d=2500]\\ \Rightarrow a_{10}=3000+(10-1)\mathrm{2500.............}[q=10]\\ \Rightarrow a_{10}=3000+9\cdot \mathrm{2500.....................}[simplify]\\ \Rightarrow a_{10}=3000+22500\\ \Rightarrow a_{10}=25500\end{array}$

Hence, the value of 10^th question is worth $25,500 .

(b)

To determine

To Calculate:

If all questions are answered correctly, how much are the winnings?

Answer to Problem 75SR

If all the questions are correct, the contestant will win $142,500 .

Explanation of Solution

Given Information :

Value of 1^st question = $3000 = first term of AP = a

From part (a) , we have the Arithematic Progression :

3000 , 3000+2500 , 3000+2(2500) , ……..

And 10^th term of the AP = 25,500 = $a_{10}$

Formula Used:

For A.P

  $a,a+d,a+2d,\mathrm{........}$

n-th term is given by:

  $\begin{array}{l}{a}_n=a+(n-1)d\mathrm{...........................}(1)\\ where\\ a=first\_term\_of\_AP\\ d=common\_difference\end{array}$

Sum of n terms is given by:

  $\begin{array}{l}{S}_n=\frac{n}{2}[2a+(n-1)d]\\ \Rightarrow S_n=\frac{n}{2}[a+a_n]\mathrm{........................}(2)\end{array}$

Calculation:

We need to find the sum of first 10 terms of the Arithematic Progression :

  $\begin{array}{l}{S}_q=\frac{q}{2}[a+a_q]\\ \Rightarrow S_{10}=\frac{10}{2}[3000+25,500]\mathrm{................}[q=10,a=3000,a_{10}=25,500]\\ \Rightarrow S_{10}=5[28,500]\mathrm{..............................}[simplify]\\ \Rightarrow S_{10}=142,500\end{array}$

So, if all the questions are answered correctly , the contestant will get $142,500.