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 (2−52).

  • A(2−52)=(2+52)+2+52+2+52+2+52=238
  • B(2−52)=(2−52)+2−52+2−52+2−52=−188
  • C(2−52)=2+52+2+52+2+52=184
  • D(2−52)=(2−52)+2−52+2−52+2−52=−178
  • E(2−52)=(2−52)+2−52+2−52=−178

Q2:

Expand and then evaluate 19𝑟.

  • A19×1+19×2+19×4=1,045.
  • B19×1+19×2+19×3+19×4=874.
  • C19×2+19×3+19×4=1,026.
  • D19×1+19×2+19×3+19×4=570.

Q3:

Expand and then evaluate (31𝑟+80).

  • A(31×1−80)+(31×2−80)+(31×3−80)+(31×4−80)=308
  • B(31×1+80)+(31×2+80)+(31×3+80)+(31×4+80)=1,051
  • C(31×1+80)+(31×2+80)+(31×3+80)+(31×4+80)=1,428
  • D(31×1+80)+(31×2+80)+(31×3+80)+(31×4+80)=1,193

Q4:

Evaluate 14(2).

Q5:

Evaluate −25.

Q6:

Evaluate −8𝑟.

  • A−143
  • B−985
  • C−263
  • D−9415

Q7:

Evaluate 12.

  • A−152
  • B2932
  • C3132
  • D−3132
  • E152

Q8:

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

  • A∞(32×𝑟)(33×𝑟)
  • B(𝑟+31)(𝑟+32)
  • C∞(𝑟+31)(𝑟+32)
  • D∞(𝑟+31)+(𝑟+32)

Q9:

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

  • A648𝑟
  • B648𝑟
  • C∞648𝑟
  • D(𝑟+648)

Q10:

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

  • A8𝑟
  • B8𝑟
  • C8𝑟
  • D∞8𝑟

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.

  • A𝑟, 10 cubes
  • B𝑟, 21 cubes
  • C𝑟, 10 cubes
  • D𝑟, 15 cubes

Q12:

Expand and then evaluate 1𝑟+42−1𝑟+41.

  • A143−142+144−143+145−144+⋯+1𝑛+42−1𝑛+41=𝑛−8442(𝑛+42)
  • B143−142+144−143+145−144+⋯+1𝑛+42−1𝑛+41=−𝑛42(𝑛+42)
  • C143−142+144−143+145−144+⋯+1𝑛+42−1𝑛+41=𝑛42(𝑛+42)
  • D143−142+144−143+145−144+⋯+1𝑛+42−1𝑛+41=𝑛+8442(𝑛+42)

Q13:

Evaluate (2).

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

Q14:

Expand −2+7𝑟.

  • A−1−98−119−1310−1511
  • B1,98,119,1310,1511
  • C1+98+119+1310+1511
  • D−1,−98,−119,−1310,−1511

Q15:

Expand and then evaluate (−63−71𝑟).

  • A(−63−71×1)+(−63−71×2)+(−63−71×3)+(−63−71×4), −962
  • B(−63−71×1)+(−63−71×2)+(−63−71×4), −686
  • C(−63+71×1)+(−63+71×2)+(−63+71×3)+(−63+71×4), −962
  • D(−63+71×1)+(−63+71×2)+(−63+71×3), 458

Q16:

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

  • A−45+96𝑟+8𝑟=−45+96×1+8×1+−45+96×2+8×2+−45+96×3+8×3+−45+96×4+8×4=841
  • B−45+96𝑟+8𝑟=−45+96×1−8×1+−45+96×2−8×2+−45+96×3−8×3+−45+96×4−8×4=540
  • C−45+96𝑟+8𝑟=−45+96×1+8×1+−45+96×2+8×2+−45+96×3+8×3+−45+96×4+8×4=1,020
  • D−45+96𝑟+8𝑟=−45+96×1+8×1+−45+96×2+8×2+−45+96×3+8×3+−45+96×4+8×4=705
  • E−45+96𝑟+8𝑟=−45−96×1+8×1+−45−96×2+8×2+−45−96×3+8×3+−45−96×4+8×4=−900

Q17:

Expand 1𝑟+16−1𝑟+18.

  • A−2323−1180−2399−1220−2483−1264
  • B−36323−110−12133−9110−12161−344
  • C36323+110+12133+9110+12161+344
  • D2323+1180+2399+1220+2483+1264

Q18:

Expand and then evaluate −5×15.

  • A−5,−1,−15,−125,−1125, −781125
  • B−5−1−15−125⋯, −15625
  • C−5−1−15−125−1125, −781125
  • D−5,−1,−15,−125,⋯, −15625

Q19:

Expand and then evaluate ((−1)−42𝑟).

  • A(−1+42×1)+(−1)+42×2+(−1)+42×3, 251
  • B(−1−42×1)+(−1)−42×2+(−1)−42×3, −170
  • C(−1−42×1)+(−1)−42×2+(−1)−42×3, −253
  • D(−1−42×1)+(−1)−42×2+(−1)−42×3, −169

Q20:

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

  • A∞(10−2)
  • B∞9
  • C(10−2)
  • D9
  • E(11−3)

Q21:

Express the series 496−497+498−499+⋯−531 in sigma notation.

  • A∞(−1)(𝑟+495)
  • B(−1)(𝑟+495)
  • C∞(−1)(𝑟+495)
  • D(−1)(𝑟+495)

Q22:

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

  • A𝑟+1
  • B𝑟+26
  • C𝑟
  • D𝑟

Q23:

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

  • A(−𝑟)
  • B𝑟
  • C∞𝑟
  • D𝑟

Q24:

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

  • A5𝑟(𝑟+1)
  • B5𝑟(𝑟+1)
  • C∞5𝑟(𝑟+1)
  • D5𝑟(𝑟+1)

Q25:

Express the series 4+5+6+7+⋯+12 in sigma notation.

  • A𝑟
  • B3𝑟
  • C3𝑟
  • D𝑟

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