(a) Find the sum of the Arithmetic Progression (AP) 1, 3, 5,..., 101. (b) Out of the 95 travellers interviewed, 7 travelled by bus and train only, 3 by trai...
Question 1 Report
(a) Find the sum of the Arithmetic Progression (AP) 1, 3, 5,..., 101.
(b) Out of the 95 travellers interviewed, 7 travelled by bus and train only, 3 by train and car only and 8 travelled by all 3 means of transport. The number, x, of travellerswho travelled by bus only was equal to the number who travelled by bus and car only. If 47 people travelled by bus and 30 by train :
(i) represent this information in a Venn diagram;
(ii) calculate the
(1) value of x ; (2) number who travelled by at least two means.
Answer Details
(a) Sum of the AP
For \(1,3,5,\ldots,101\):
\(a=1\), \(d=2\), and \(l=101\).
\[ 101=1+(n-1)2 \] \[ 100=2(n-1)\Rightarrow n-1=50\Rightarrow n=51 \] \[ S_n=\frac{n}{2}(a+l) =\frac{51}{2}(1+101) =\frac{51}{2}(102) =2601 \]The sum is \(2601\).
(b)(i) Venn diagram
Let \(B\), \(T\), and \(C\) represent travellers who used bus, train, and car respectively. Since bus only and bus-and-car only are both \(x\), use the bus total:
\[ x+7+x+8=47 \] \[ 2x+15=47\Rightarrow 2x=32\Rightarrow x=16 \]The train-only region is:
\[ 30-(7+3+8)=12 \]As all \(95\) travellers are represented in the diagram, the car-only region is:
\[ 95-(16+12+16+7+3+8)=33 \](ii)(1) The value of \(x\) is \(16\).
(ii)(2) Travellers who used at least two means of transport
“At least two” includes each region for exactly two means, as well as the region for all three means:
\[ 7+16+3+8=34 \]\(34\) travellers travelled by at least two means of transport.
Examination reminder: When finding “at least two” in a Venn diagram, include the central intersection because those people used all three means, which is also at least two.
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