Algebra, Sequences, Complete/Continue the Sequence

These problems provide the first few terms of a sequence and ask to find the next term(s) or to identify the underlying pattern or rule. This tests pattern recognition skills for various types of sequences (arithmetic, geometric, quadratic, Fibonacci-like, visual, etc.).

• Complete the sequence

  Complete the missing number:

  `2, 3, 5, 9, ?, 33`

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  Topics:

  Arithmetic Algebra -> Sequences -> Complete/Continue the Sequence

• Continue the Sequence

  `2, 6, 12, 20, 30…`

  a. What is the next number (the sixth) in the sequence?

  b. What is the `2016`th number in the sequence?

  Topics:

  Arithmetic Logic -> Reasoning / Logic Algebra -> Sequences -> Arithmetic Progression / Arithmetic Sequence Algebra -> Sequences -> Complete/Continue the Sequence
  Sources:
  • problems.ru web site

• Question

  `120` identical spheres are arranged in the shape of a triangular pyramid. How many layers are there in the pyramid?

  Note: This is a pyramid, which is a three-dimensional shape, and not a triangle in a plane.

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  Topics:

  Geometry -> Solid Geometry / Geometry in Space Arithmetic Logic -> Reasoning / Logic Proof and Example -> Constructing an Example / Counterexample Algebra -> Sequences -> Arithmetic Progression / Arithmetic Sequence Algebra -> Sequences -> Complete/Continue the Sequence Number Theory -> Triangular Numbers