Remove multiples and sum a bouncing path · source reference

Printed question

(a) Find the sum of all integers in $\{1,2,3,\ldots,2014\}$ that are not multiples of 3. [4]

(b) An elastic ball released from height $h$ falls vertically onto a horizontal floor and rebounds to height $ah$, $0<a<1$. Initially it is released from height $H$.

(i) Write an expression in $H,a,n$ for its height after bounce $n$. [1]

Given $a=\frac34$,

(ii) find the number of bounces before the height is less than $H/20$ for the first time [3];

(iii) calculate in terms of $H$ the total distance just before it comes to complete rest. [2]

Printed working

(a) Multiples of 3 are $3(1),3(2),\ldots,3(n)$ with $3n\le2014\Rightarrow n\le671\frac13$. There are 671 multiples.

Their sum $=\frac{671}2(3+2013)=676368$. Required sum $=\frac{2014}2(1+2014)-676368=1352737$.

(b)(i) Height after $n$th bounce $=a^nH$.

(ii) Given $a=3/4$, consider

$$(3/4)^nH<H/20\Rightarrow(3/4)^n<1/20\quad(H>0),$$ $$n\ln(3/4)<\ln(1/20)\Rightarrow n>\frac{\ln(1/20)}{\ln(3/4)}>10.4.$$

Least $n=11$, so 11 bounces.

(iii) Total distance

$$\begin{aligned}H+2[(3/4)H+(3/4)^2H+(3/4)^3H+\cdots]&=H+2(3H/4)[1+3/4+(3/4)^2+\cdots]\\&=H+\frac32H\frac1{1-3/4}=7H.\end{aligned}$$

Teacher notes

AP sum $S_n=\frac n2(u_1+u_n)$ with $n=671,u_1=3,u_n=3(671)$.

Required integer sum = sum of all integers in $\{1,2,\ldots,2014\}$ − sum of its multiples of 3.

Change inequality sign when dividing/multiplying by a negative value: $\ln(3/4)<0$.

Source diagrams redrawn

Printed answer key

(a) 1352737; (b)(i) $a^nH$; (ii) 11; (iii) $7H$

Mathematical review and qualifications

(a) There are 671 multiples of 3. The required sum is $2014\cdot2015/2-3\cdot671\cdot672/2=1352737$.

(b)(i) Rebound height is $a^nH$. (ii) $(3/4)^n<1/20$ gives $n>\ln(1/20)/\ln(3/4)\approx10.413$, so 11 bounces. (iii) Distance is $H+2H\sum_{n=1}^{\infty}(3/4)^n=H+2H(3)=7H$.

Provenance

2025 TJC H2 Math (9758) JC2 MSM: ARITHMETIC & GEOMETRIC PROGRESSIONS. [2014/NYJC/JC2/mid-year/P2/4] Question PDF pages 4; solution PDF pages 20, 21. The original files remain private QA records outside this native content.