Algebra, Algebraic Techniques, Telescoping Sums

A telescoping sum (or series) is one where most terms cancel out with preceding or succeeding terms, leaving only a few. Questions involve recognizing this structure (often after partial fraction decomposition or other manipulation) to easily calculate the sum.

• Question

  Calculate the sum:

  `1/(1*2)+1/(2*3)+1/(3*4)+...+1/(99*100)`

  Topics:

  Algebra -> Sequences Arithmetic -> Fractions Algebra -> Algebraic Techniques -> Telescoping Sums

• Question

  Calculate the product:

  `(1-1/2)(1-1/3)(1-1/4)*...*(1-1/100)`

  Topics:

  Algebra -> Sequences Arithmetic -> Fractions Algebra -> Algebraic Techniques -> Telescoping Sums

• Question

  Calculate the product:

  `(1-1/4)(1-1/9)(1-1/16)*...*(1-1/225)`

  Topics:

  Algebra -> Algebraic Techniques -> Short Multiplication Formulas / Algebraic Identities Algebra -> Sequences Arithmetic -> Fractions Algebra -> Algebraic Techniques -> Telescoping Sums

• Question

  Calculate the value of the expression (without a calculator):

  `(1+1/(2^2-1))(1+1/(3^2-1))(1+1/(4^2-1))*...*(1+1/(99^2-1))`

  Topics:

  Arithmetic Algebra -> Sequences Algebra -> Algebraic Techniques -> Telescoping Sums

• Question

  Calculate the value of the expression (without a calculator):

  `(2^3-1)/(2^3+1)*(3^3-1)/(3^3+1)*(4^3-1)/(4^3+1)*...*(100^3-1)/(100^3+1)`

  Topics:

  Algebra -> Algebraic Techniques -> Short Multiplication Formulas / Algebraic Identities Algebra -> Sequences Algebra -> Algebraic Techniques -> Telescoping Sums

• Question

  Simplify the expression:

  `1/(1-a)+1/(1+a)+2/(1+a^2)+4/(1+a^4)+8/(1+a^8)+16/(1+a^16)`

  Topics:

  Algebra -> Algebraic Techniques -> Short Multiplication Formulas / Algebraic Identities Algebra -> Equations Algebra -> Algebraic Techniques -> Telescoping Sums

• 50 to the Power of

  Show that in the rightmost 504 digits of `1+50+50^2+...+50^1000`

  Each digit appears a number of times divisible by 12

  Show hint

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Algebra -> Sequences -> Arithmetic Progression / Arithmetic Sequence Algebra -> Algebraic Techniques -> Telescoping Sums
  Sources:
  • Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 3 Question 2

• Partitioned Roots

  Find the sum `1/{sqrt1+sqrt2}+1/{sqrt2+sqrt3}+...+1/{sqrt99+sqrt100}`.

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  Topics:

  Algebra -> Algebraic Techniques -> Telescoping Sums Algebra -> Algebraic Techniques -> Roots / Radicals
  Sources:
  • Junior Science Team Exercises, 2024-2025, Assignment 1 Question 2