Worksheet: Sigma Notation

In this worksheet, we will practice expressing a series in sigma notation and how to expand and evaluate the sigma notation.

Q1:

Expand and then evaluate (252).

  • A ( 2 5 2 ) = ( 2 5 2 ) + 2 5 2 + 2 5 2 + 2 5 2 = 1 8 8
  • B ( 2 5 2 ) = ( 2 + 5 2 ) + 2 + 5 2 + 2 + 5 2 + 2 + 5 2 = 2 3 8
  • C ( 2 5 2 ) = ( 2 5 2 ) + 2 5 2 + 2 5 2 = 1 7 8
  • D ( 2 5 2 ) = ( 2 5 2 ) + 2 5 2 + 2 5 2 + 2 5 2 = 1 7 8
  • E ( 2 5 2 ) = 2 + 5 2 + 2 + 5 2 + 2 + 5 2 = 1 8 4

Q2:

Expand and then evaluate 19𝑟.

  • A 1 9 × 1 + 1 9 × 2 + 1 9 × 4 = 1 , 0 4 5 .
  • B 1 9 × 1 + 1 9 × 2 + 1 9 × 3 + 1 9 × 4 = 8 7 4 .
  • C 1 9 × 2 + 1 9 × 3 + 1 9 × 4 = 1 , 0 2 6 .
  • D 1 9 × 1 + 1 9 × 2 + 1 9 × 3 + 1 9 × 4 = 5 7 0 .

Q3:

Expand and then evaluate (31𝑟+80).

  • A ( 3 1 × 1 8 0 ) + ( 3 1 × 2 8 0 ) + ( 3 1 × 3 8 0 ) + ( 3 1 × 4 8 0 ) = 3 0 8
  • B ( 3 1 × 1 + 8 0 ) + ( 3 1 × 2 + 8 0 ) + ( 3 1 × 3 + 8 0 ) + ( 3 1 × 4 + 8 0 ) = 1 , 0 5 1
  • C ( 3 1 × 1 + 8 0 ) + ( 3 1 × 2 + 8 0 ) + ( 3 1 × 3 + 8 0 ) + ( 3 1 × 4 + 8 0 ) = 1 , 4 2 8
  • D ( 3 1 × 1 + 8 0 ) + ( 3 1 × 2 + 8 0 ) + ( 3 1 × 3 + 8 0 ) + ( 3 1 × 4 + 8 0 ) = 1 , 1 9 3

Q4:

Evaluate 14(2).

  • A62
  • B 6 3 4
  • C 2
  • D126

Q5:

Evaluate 25.

Q6:

Evaluate 8𝑟.

  • A 1 4 3
  • B 9 8 5
  • C 2 6 3
  • D 9 4 1 5

Q7:

Evaluate 12.

  • A 1 5 2
  • B 2 9 3 2
  • C 3 1 3 2
  • D 3 1 3 2
  • E 1 5 2

Q8:

Express the series (32×33)+(33×34)+(34×35)+ in sigma notation.

  • A ( 3 2 × 𝑟 ) ( 3 3 × 𝑟 )
  • B ( 𝑟 + 3 1 ) ( 𝑟 + 3 2 )
  • C ( 𝑟 + 3 1 ) ( 𝑟 + 3 2 )
  • D ( 𝑟 + 3 1 ) + ( 𝑟 + 3 2 )

Q9:

Express the series 54×12+54×24+54×36++54×240 in sigma notation.

  • A 6 4 8 𝑟
  • B 6 4 8 𝑟
  • C 6 4 8 𝑟
  • D ( 𝑟 + 6 4 8 )

Q10:

Express the series 8+32+72+128++512 in sigma notation.

  • A 8 𝑟
  • B 8 𝑟
  • C 8 𝑟
  • D 8 𝑟

Q11:

A child wants to build a pyramid using identical wooden cubes. The pyramid’s apex consists of one cube, its second row consists of two cubes, and its third row consists of three cubes, and so on for five rows. Use the sigma notation to describe the amount of cubes needed to build the pyramid and then calculate this number.

  • A 𝑟 , 10 cubes
  • B 𝑟 , 21 cubes
  • C 𝑟 , 10 cubes
  • D 𝑟 , 15 cubes

Q12:

Expand and then evaluate 1𝑟+421𝑟+41.

  • A 1 4 3 1 4 2 + 1 4 4 1 4 3 + 1 4 5 1 4 4 + + 1 𝑛 + 4 2 1 𝑛 + 4 1 = 𝑛 8 4 4 2 ( 𝑛 + 4 2 )
  • B 1 4 3 1 4 2 + 1 4 4 1 4 3 + 1 4 5 1 4 4 + + 1 𝑛 + 4 2 1 𝑛 + 4 1 = 𝑛 4 2 ( 𝑛 + 4 2 )
  • C 1 4 3 1 4 2 + 1 4 4 1 4 3 + 1 4 5 1 4 4 + + 1 𝑛 + 4 2 1 𝑛 + 4 1 = 𝑛 4 2 ( 𝑛 + 4 2 )
  • D 1 4 3 1 4 2 + 1 4 4 1 4 3 + 1 4 5 1 4 4 + + 1 𝑛 + 4 2 1 𝑛 + 4 1 = 𝑛 + 8 4 4 2 ( 𝑛 + 4 2 )

Q13:

Evaluate (2).

  • A8,064
  • B26,637
  • C12,672
  • D10,560

Q14:

Expand 2+7𝑟.

  • A 1 9 8 1 1 9 1 3 1 0 1 5 1 1
  • B 1 , 9 8 , 1 1 9 , 1 3 1 0 , 1 5 1 1
  • C 1 + 9 8 + 1 1 9 + 1 3 1 0 + 1 5 1 1
  • D 1 , 9 8 , 1 1 9 , 1 3 1 0 , 1 5 1 1

Q15:

Expand and then evaluate (6371𝑟).

  • A ( 6 3 7 1 × 1 ) + ( 6 3 7 1 × 2 ) + ( 6 3 7 1 × 3 ) + ( 6 3 7 1 × 4 ) , 9 6 2
  • B ( 6 3 7 1 × 1 ) + ( 6 3 7 1 × 2 ) + ( 6 3 7 1 × 4 ) , 6 8 6
  • C ( 6 3 + 7 1 × 1 ) + ( 6 3 + 7 1 × 2 ) + ( 6 3 + 7 1 × 3 ) + ( 6 3 + 7 1 × 4 ) , 9 6 2
  • D ( 6 3 + 7 1 × 1 ) + ( 6 3 + 7 1 × 2 ) + ( 6 3 + 7 1 × 3 ) , 458

Q16:

Expand and then use direct substitution to evaluate 45+96𝑟+8𝑟.

  • A 4 5 + 9 6 𝑟 + 8 𝑟 = 4 5 + 9 6 × 1 + 8 × 1 + 4 5 + 9 6 × 2 + 8 × 2 + 4 5 + 9 6 × 3 + 8 × 3 + 4 5 + 9 6 × 4 + 8 × 4 = 8 4 1
  • B 4 5 + 9 6 𝑟 + 8 𝑟 = 4 5 + 9 6 × 1 8 × 1 + 4 5 + 9 6 × 2 8 × 2 + 4 5 + 9 6 × 3 8 × 3 + 4 5 + 9 6 × 4 8 × 4 = 5 4 0
  • C 4 5 + 9 6 𝑟 + 8 𝑟 = 4 5 + 9 6 × 1 + 8 × 1 + 4 5 + 9 6 × 2 + 8 × 2 + 4 5 + 9 6 × 3 + 8 × 3 + 4 5 + 9 6 × 4 + 8 × 4 = 1 , 0 2 0
  • D 4 5 + 9 6 𝑟 + 8 𝑟 = 4 5 + 9 6 × 1 + 8 × 1 + 4 5 + 9 6 × 2 + 8 × 2 + 4 5 + 9 6 × 3 + 8 × 3 + 4 5 + 9 6 × 4 + 8 × 4 = 7 0 5
  • E 4 5 + 9 6 𝑟 + 8 𝑟 = 4 5 9 6 × 1 + 8 × 1 + 4 5 9 6 × 2 + 8 × 2 + 4 5 9 6 × 3 + 8 × 3 + 4 5 9 6 × 4 + 8 × 4 = 9 0 0

Q17:

Expand 1𝑟+161𝑟+18.

  • A 2 3 2 3 1 1 8 0 2 3 9 9 1 2 2 0 2 4 8 3 1 2 6 4
  • B 3 6 3 2 3 1 1 0 1 2 1 3 3 9 1 1 0 1 2 1 6 1 3 4 4
  • C 3 6 3 2 3 + 1 1 0 + 1 2 1 3 3 + 9 1 1 0 + 1 2 1 6 1 + 3 4 4
  • D 2 3 2 3 + 1 1 8 0 + 2 3 9 9 + 1 2 2 0 + 2 4 8 3 + 1 2 6 4

Q18:

Expand and then evaluate 5×15.

  • A 5 , 1 , 1 5 , 1 2 5 , 1 1 2 5 , 7 8 1 1 2 5
  • B 5 1 1 5 1 2 5 , 1 5 6 2 5
  • C 5 1 1 5 1 2 5 1 1 2 5 , 7 8 1 1 2 5
  • D 5 , 1 , 1 5 , 1 2 5 , , 1 5 6 2 5

Q19:

Expand and then evaluate ((1)42𝑟).

  • A ( 1 + 4 2 × 1 ) + ( 1 ) + 4 2 × 2 + ( 1 ) + 4 2 × 3 , 251
  • B ( 1 4 2 × 1 ) + ( 1 ) 4 2 × 2 + ( 1 ) 4 2 × 3 , 1 7 0
  • C ( 1 4 2 × 1 ) + ( 1 ) 4 2 × 2 + ( 1 ) 4 2 × 3 , 2 5 3
  • D ( 1 4 2 × 1 ) + ( 1 ) 4 2 × 2 + ( 1 ) 4 2 × 3 , 1 6 9

Q20:

Express the series 8+98+998+9,998+ in sigma notation to 𝑛 terms.

  • A ( 1 0 2 )
  • B 9
  • C ( 1 0 2 )
  • D 9
  • E ( 1 1 3 )

Q21:

Find the value of 7+9+11++3712+14+16++172.

  • A 7 3 6 3 3
  • B 8 8 1 , 8 6 3
  • C 1 , 8 6 3 8 8
  • D 1 1 4 6
  • E 3 3 7 3 6

Q22:

Express the series 496497+498499+531 in sigma notation.

  • A ( 1 ) ( 𝑟 + 4 9 5 )
  • B ( 1 ) ( 𝑟 + 4 9 5 )
  • C ( 1 ) ( 𝑟 + 4 9 5 )
  • D ( 1 ) ( 𝑟 + 4 9 5 )

Q23:

Express the series 26+27+28+29+30++84 in sigma notation .

  • A 𝑟 + 1
  • B 𝑟 + 2 6
  • C 𝑟
  • D 𝑟

Q24:

Express the series 125+216+343++729 in sigma notation.

  • A ( 𝑟 )
  • B 𝑟
  • C 𝑟
  • D 𝑟

Q25:

Express the series 526×27+527×28+528×29+529×30++534×35 in sigma notation.

  • A 5 𝑟 ( 𝑟 + 1 )
  • B 5 𝑟 ( 𝑟 + 1 )
  • C 5 𝑟 ( 𝑟 + 1 )
  • D 5 𝑟 ( 𝑟 + 1 )

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