4. Write neatly and legibly. QUESTION 1: TRANSPORT Mr. Dlamini transport people between Butterworth and East London using a bus which has a capacity of 100 people. The transport charge starts with a minimum charge of R8 and thereafter it was increased by R2 for each kilometre. On a particular day the bus was full with passengers from Butterworth. In each and every kilometre there was a passenger getting off while no new passenger entered in the bus. Mr. Dlamini was calculating total money in his hand after each passenger was getting out. The total amount he has after each stop can be written as a series as follows: $$ S_{n}=8+10+12+14+\ldots $$ Use the series to calculate the following questions (i) Term $$ 2\left(T_{2}\right)= $$ Sum of first two terms $$ \left(\mathrm{S}_{2}\right)= $$ $$ \qquad $$ (ii) Express $$ T_{2} $$ in terms of $$ S_{1} $$ and $$ S_{2} $$. $$ \qquad $$ 1.2 (i) Term $$ 3\left(T_{3}\right)= $$ $$ \qquad $$ Sum of first three terms $$ \left(\mathrm{S}_{3}\right)= $$. $$ \qquad $$ (ii) Express $$ T_{3} $$ in terms of $$ S_{3} $$ and $$ S_{2} $$. $$ \qquad $$ (i) $$ \operatorname{Term} 4\left(T_{4}\right)= $$ $$ \qquad $$ Sum of first four terms $$ \left(S_{4}\right)= $$. $$ \qquad $$ (ii) Express $$ T_{4} $$ in terms of $$ S_{3} $$ and $$ S_{4} $$.

Question

UpStudy AI · Accepted Answer

We are given the series of amounts that Mr. Dlamini collects after each kilometre, which is written as:

Sₙ = 8 + 10 + 12 + 14 + …

This is an arithmetic series where the first term (a) is 8 and the common difference (d) is 2. The nth term is given by:

Tₙ = a + (n – 1)d.

Also, the sum of the first n terms of an arithmetic series is:

Sₙ = (n/2) [2a + (n – 1)d].

We now calculate the values step by step:

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1. For n = 2:
────────────────────────────

(i) Calculate Term 2 (T₂):

T₂ = 8 + (2 – 1) × 2  
     = 8 + 2  
     = 10

Calculate the sum of the first 2 terms (S₂):

S₂ = 8 + 10  
     = 18

(ii) Express T₂ in terms of S₁ and S₂:

Note that S₁ = T₁ = 8. Therefore,
  T₂ = S₂ – S₁  
     = 18 – 8  
     = 10

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2. For n = 3:
────────────────────────────

(i) Calculate Term 3 (T₃):

T₃ = 8 + (3 – 1) × 2  
     = 8 + 4  
     = 12

Calculate the sum of the first 3 terms (S₃):

S₃ = 8 + 10 + 12  
     = 30

(ii) Express T₃ in terms of S₃ and S₂:

Since S₃ = S₂ + T₃, we have:
  T₃ = S₃ – S₂  
     = 30 – 18  
     = 12

────────────────────────────
3. For n = 4:
────────────────────────────

(i) Calculate Term 4 (T₄):

T₄ = 8 + (4 – 1) × 2  
     = 8 + 6  
     = 14

Calculate the sum of the first 4 terms (S₄):

S₄ = 8 + 10 + 12 + 14  
     = 44

(ii) Express T₄ in terms of S₄ and S₃:

Since S₄ = S₃ + T₄, we get:
  T₄ = S₄ – S₃  
     = 44 – 30  
     = 14

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Summary of Answers:
────────────────────────────

1. For n = 2:
  (i) Term 2, T₂ = 10  
     Sum of first two terms, S₂ = 18  
  (ii) T₂ = S₂ – S₁

2. For n = 3:
  (i) Term 3, T₃ = 12  
     Sum of first three terms, S₃ = 30  
  (ii) T₃ = S₃ – S₂

3. For n = 4:
  (i) Term 4, T₄ = 14  
     Sum of first four terms, S₄ = 44  
  (ii) T₄ = S₄ – S₃

This completes the required calculations and expressions.
Term 2 (T₂) = 10 Sum of first two terms (S₂) = 18 T₂ = S₂ – S₁ Term 3 (T₃) = 12 Sum of first three terms (S₃) = 30 T₃ = S₃ – S₂ Term 4 (T₄) = 14 Sum of first four terms (S₄) = 44 T₄ = S₄ – S₃

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