Problem Solving Arithmetic Progression and Geometric Progression

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Problem SolvingArithmetic Progressionand
Geometric Progression
Last Updated October 11, 2005
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Review -- Arithmetic
Sum of n terms
nth term
Jeff Bivin -- LZHS
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Review -- Geometric
Sum of n terms
nth term
an a1r(n-1)
Jeff Bivin -- LZHS
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Example 1

• The sum of the first n terms of a progression
• is given by Sn n2 3n. Find, in terms of n
• the nth term.
• Sn n2 3n
• Sn-1 (n-1)2 3(n-1)
• n2 2n 1 3n 3
• n2 n 2
• Tn Sn Sn 1
• n2 3n (n2 n 2)
• 2n 2

Jeff Bivin -- LZHS
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SPM ANALYSIS
YEAR PROGRESSION TOTAL MARKS
2004 GP 8 MARKS
2005 AP 6 MARKS
2006 AP 7 MARKS
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Year 2004, SPM PAPER 2

• The diagram shows the arrangement of the first 3
  of an
• infinite series of similar triangles. The first
  triangle has a
• base of x cm and a height of y cm. The
  measurements of the
• base and the height of each subsequent triangle
  are half of
• the measurement of its previous one.
• Show that the areas of the triangles form a
  geometric
• progression and state the common ratio.
  3 marks
• Given that x 80 and y 40 5 marks
• (i) determine which triangle has an area of
  6.25 cm2
• (ii) find the sum to infinity of the areas
  in cm2 of the triangles

Jeff Bivin -- LZHS
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SOLUTION (a)

• Area of 1st triangle A1
• Area of 2nd triangle A2
• Area of 3rd triangle A3
• The area of triangles form a GP with common ratio
  

Jeff Bivin -- LZHS
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SOLUTION (b)(i)

• Area of nth triangle An A1r n 1
• 6.25

?The fifth triangle has area of 6.25 cm2
Jeff Bivin -- LZHS
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SOLUTION (b)(ii)

• Sum to infinity

cm2
Jeff Bivin -- LZHS
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Year 2005, SPM PAPER 2

• Diagram above shows part of an arrangement of
  bricks of
• equal size.
• The number of bricks in the lowest row is 100.
  For each of
• the other rows, the number of bricks is 2 less
  than in the row
• below. The height of each brick is 6 cm.
• Ali builds a wall by arranging bricks in this
  way.
• The number of bricks in the highest row is 4.
  Calculate
• The height, in cm, of the wall 3 marks
• The total price of the bricks used if the price
  of
• of one brick is 40 sen. 3 marks

Jeff Bivin -- LZHS
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Year 2005, SPM PAPER 2

• The arrangement of bricks is in arithmetic
  progression
• 100, 98, 96, 94, 92, .8, 6, 4.
• a 100, d -2
• Let Tn 4
• 100 (n 1)(-2) 4
• 96 2(n 1)
• 48 n - 1
• n 49
• The height of the wall 49 ? 6 294 cm.
• Total bricks used

The total price of bricks used 2, 548 ? 0.40

RM1, 019.20
Jeff Bivin -- LZHS
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Year 2006, SPM PAPER 2

• Two companies, Delta and Omega, start to sell
  cars at the same
• time.
• Delta sells k cars in the first month and its
  sales increase constantly by m cars every
  subsequent month. It sells 240 cars in the 8th
  month and the total sales for the first 10 months
  are 1900 cars.
• Find the value of k and of m. 5 marks
• Omega sells 80 cars in the first month and its
  sales increase constantly by 22 cars every
  subsequent month.
• If both companies sell the same number of
  cars in the nth month, find the value of n.
• 2 marks

Jeff Bivin -- LZHS
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• k, km, k2m, k 3m, k4m,..
• 240 k 7m..(1)
• 1900 5(2k9m)
• 380 2k 9m.(2)
• ? 2 (2) 480 380 14m 9m
• 100 5m
• m 20
• Substitute m 20 into (1)
• 240 k 7(20)
• k 240 140
• k 100

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• Delta 100, 120, 140, 160, 180, .
• Tn 100 20(n 1)
• Omega 80, 102, 124, 146, 168,
• Tn 80 22(n 1)
• 100 20(n 1) 80 22(n 1)
• 100 20n 20 80 22n 22
• 80 20n 58 22n
• 22 2n
• n 11