Algebra, Sequences

A sequence is an ordered list of numbers (or other items) that often follows a specific rule or pattern. This topic covers identifying patterns, finding specific terms, determining general formulas (`n`-th term), and understanding different types of sequences (arithmetic, geometric, recursive).

• Question

  The following numbers are written on the board:

  `1,2^1,2^2,2^3,2^4,2^5`

  In one operation, you are allowed to erase two numbers written on the board and write their (non-negative) difference in their place.

  Is it possible to reach a state, through such operations, where only the number `15` is written on the board?

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  Topics:

  Arithmetic Proof and Example -> Constructing an Example / Counterexample Algebra -> Sequences
  Sources:
  • problems.ru web site

• Question

  The sum of several numbers is equal to `1`. Is it possible that the sum of their squares is less than one-tenth?

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  Topics:

  Proof and Example -> Constructing an Example / Counterexample Algebra -> Sequences Arithmetic -> Fractions Algebra -> Inequalities -> Averages / Means Minimum and Maximum Problems / Optimization Problems
  Sources:
  • problems.ru web site

• Question

  Prove that any real number can be written as the sum of 9 numbers, each of which is composed only of the digits 0 and 7.

  Topics:

  Number Theory Arithmetic Proof and Example Algebra -> Sequences
  Sources:
  • Tournament of Towns, 1980-1981, Spring, Main Version, Grades 11-12 Question 3

• Question

  A `29×29` table contains all integers from `1` to `29`, each appearing exactly `29` times. The sum of all numbers above the main diagonal is exactly three times greater than the sum of all numbers below the main diagonal. What number is written in the central cell of the table?

  Topics:

  Arithmetic Logic -> Reasoning / Logic Algebra -> Sequences Algebra -> Inequalities -> Averages / Means Combinatorics -> Number Tables Minimum and Maximum Problems / Optimization Problems

• Bacteria in a Test Tube

  A scientist has a test tube containing bacteria. Every second, each bacterium splits into two. After two hours, the test tube was completely full of bacteria. How long before that was the test tube exactly half full?

  Topics:

  Arithmetic Logic -> Reasoning / Logic Algebra -> Sequences Algebra -> Word Problems -> Solving Word Problems "From the End" / Working Backwards

• 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