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Formulas: sequences and induction, grade 11 (5 units)

A sequence and the sum of its first terms

  • The term from the sum – Sn\displaystyle S_n the sum of the first n\displaystyle n terms
    an\displaystyle a_n =\displaystyle = Sn−Sn−1\displaystyle S_n-S_{n-1}n≥2\displaystyle n\geq2
  • The first term is the sum of one term
    a1\displaystyle a_1 =\displaystyle = S1\displaystyle S_1

The arithmetic sequence

  • The recursion rule – d\displaystyle d the common difference
    an+1=an+d\displaystyle a_{n+1}=a_n+d
  • The general term
    an\displaystyle a_n =\displaystyle = a1+(n−1)d\displaystyle a_1+(n-1)d
  • The arithmetic mean of the neighbours
    an\displaystyle a_n =\displaystyle = an−1+an+12\displaystyle \frac{a_{n-1}+a_{n+1}}{2}n≥2\displaystyle n\geq2
  • The sum of the first n\displaystyle n terms
    Sn\displaystyle S_n =\displaystyle = n(a1+an)2\displaystyle \frac{n(a_1+a_n)}{2}
  • The sum without the last term
    Sn\displaystyle S_n =\displaystyle = n[2a1+(n−1)d]2\displaystyle \frac{n\left[2a_1+(n-1)d\right]}{2}

The geometric sequence

  • The recursion rule – q\displaystyle q the common ratio
    an+1=an⋅q\displaystyle a_{n+1}=a_n\cdot q
  • The general term
    an\displaystyle a_n =\displaystyle = a1⋅qn−1\displaystyle a_1\cdot q^{n-1}
  • The geometric mean: a term’s square is the product of its neighbours
    an2\displaystyle a_n^2 =\displaystyle = an−1⋅an+1\displaystyle a_{n-1}\cdot a_{n+1}n≥2\displaystyle n\geq2
  • The sum of the first n\displaystyle n terms
    Sn\displaystyle S_n =\displaystyle = a1(qn−1)q−1\displaystyle \frac{a_1(q^n-1)}{q-1}q≠1\displaystyle q\neq1

The infinite geometric sequence

  • The sum of a convergent sequence
    S\displaystyle S =\displaystyle = a11−q\displaystyle \frac{a_1}{1-q}−1<q<1, q≠0\displaystyle -1<q<1,\ q\neq0

The laws of powers

  • A product of powers of one base
    am⋅an\displaystyle a^m\cdot a^n =\displaystyle = am+n\displaystyle a^{m+n}
  • A quotient of powers of one base
    aman\displaystyle \frac{a^m}{a^n} =\displaystyle = am−n\displaystyle a^{m-n}a≠0\displaystyle a\neq0
  • A power of a power
    (am)n\displaystyle \left(a^m\right)^n =\displaystyle = amn\displaystyle a^{mn}
  • A power with exponent 0
    a0\displaystyle a^0 =\displaystyle = 1\displaystyle 1a≠0\displaystyle a\neq0
  • A power with a negative exponent
    a−n\displaystyle a^{-n} =\displaystyle = 1an\displaystyle \frac{1}{a^n}a≠0\displaystyle a\neq0

Sums proved by induction

  • The sum of the natural numbers
    1+2+3+…+n\displaystyle 1+2+3+\ldots+n =\displaystyle = n(n+1)2\displaystyle \frac{n(n+1)}{2}
  • The sum of the odd numbers
    1+3+5+…+(2n−1)\displaystyle 1+3+5+\ldots+(2n-1) =\displaystyle = n2\displaystyle n^2
  • The sum of the squares
    12+22+32+…+n2\displaystyle 1^2+2^2+3^2+\ldots+n^2 =\displaystyle = n(n+1)(2n+1)6\displaystyle \frac{n(n+1)(2n+1)}{6}
  • The sum of the cubes
    13+23+33+…+n3\displaystyle 1^3+2^3+3^3+\ldots+n^3 =\displaystyle = n2(n+1)24\displaystyle \frac{n^2(n+1)^2}{4}