Comprehensive Study Guide for Arithmetic and Geometric Sequences

Principles of Arithmetic Sequences

  • Definition: A sequence (un)(u_n) is called arithmetic with a raison rr (where r∈Rr \in \mathbb{R}) if, for all n∈Nn \in \mathbb{N}, the following relationship holds:     un+1=un+ru_{n+1} = u_n + r

  • Method of Proof: To demonstrate that a sequence is arithmetic, one must prove that the difference un+1−unu_{n+1} - u_n is a constant value independent of nn.

  • General Term Expressions:

    • If the first term is u0u_0:         un=u0+n×ru_n = u_0 + n \times r
    • If the first term is u1u_1:         un=u1+(n−1)×ru_n = u_1 + (n - 1) \times r
    • Relationship between any two terms unu_n and upu_p (where n>pn > p):         un=up+(n−p)×ru_n = u_p + (n - p) \times r

  • Summation of Arithmetic Terms:

    • Sum of terms from u0u_0 to unu_n:         u0+u1+⋯+un=(n+1)×u0+un2u_0 + u_1 + \dots + u_n = (n + 1) \times \frac{u_0 + u_n}{2}
    • Generalized Summation Formula:         Sum=number of terms×first term+last term2\text{Sum} = \text{number of terms} \times \frac{\text{first term} + \text{last term}}{2}
    • Sum from index k=nk = n to mm:         Sm=∑k=nmuk=(m−n+1)×(un+um)2S_m = \sum_{k=n}^{m} u_k = \frac{(m - n + 1) \times (u_n + u_m)}{2}

  • Limits of Arithmetic Sequences:

    • If r=0r = 0: lim⁡n→+∞un=u0\lim_{n \to +\infty} u_n = u_0
    • If r>0r > 0: lim⁡n→+∞un=+∞\lim_{n \to +\infty} u_n = +\infty
    • If r<0r < 0: lim⁡n→+∞un=−∞\lim_{n \to +\infty} u_n = -\infty

Principles of Geometric Sequences

  • Definition: A sequence (un)(u_n) is called geometric with a raison qq (where q∈Rq \in \mathbb{R}) if, for all n∈Nn \in \mathbb{N}, the following relationship holds:     un+1=q×unu_{n+1} = q \times u_n

  • Method of Proof: Start with the expression for un+1u_{n+1} and attempt to write it solely as a function of unu_n, or divide un+1u_{n+1} by unu_n to find the constant raison qq.

  • General Term Expressions:

    • If the first term is u0u_0:         un=u0×qnu_n = u_0 \times q^n
    • If the first term is u1u_1:         un=u1×qn−1u_n = u_1 \times q^{n-1}
    • Relationship between any two terms unu_n and upu_p (where n>pn > p):         un=up×qn−pu_n = u_p \times q^{n-p}

  • Summation of Geometric Terms:

    • General Summation Formula where q≠1q \neq 1:         Sum=first term×1−qnumber of terms1−q\text{Sum} = \text{first term} \times \frac{1 - q^{\text{number of terms}}}{1 - q}
    • Formatted for a sum from u0u_0 to unu_n:         S=u0×1−qn+11−qS = u_0 \times \frac{1 - q^{n+1}}{1 - q}
    • Specific case for index k=nk = n to mm:         Sm=un×1−qm−n+11−qS_m = u_n \times \frac{1 - q^{m-n+1}}{1 - q}
    • Sum of powers: 1+q+q2+⋯+qn=1−qn+11−q1 + q + q^2 + \dots + q^n = \frac{1 - q^{n+1}}{1 - q}
    • If q=1q = 1, the sum of the first terms is: first term×number of terms\text{first term} \times \text{number of terms}.

  • Limits of Geometric Sequences:

    • If q=1q = 1: lim⁡n→+∞un=u0\lim_{n \to +\infty} u_n = u_0
    • If −1<q<1-1 < q < 1: lim⁡n→+∞un=0\lim_{n \to +\infty} u_n = 0
    • If q>1q > 1 and u0>0u_0 > 0: lim⁡n→+∞un=+∞\lim_{n \to +\infty} u_n = +\infty
    • If q>1q > 1 and u0<0u_0 < 0: lim⁡n→+∞un=−∞\lim_{n \to +\infty} u_n = -\infty
    • If q≤−1q \leq -1: The sequence does not admit a limit.

Sequence Characteristics and Variation

  • Boundedness:

    • Majorée (Bounded Above): A sequence is majorée if there exists a real MM such that ∀n∈I,un≤M\forall n \in I, u_n \leq M.
    • Minorée (Bounded Below): A sequence is minorée if there exists a real mm such that ∀n∈I,un≥m\forall n \in I, u_n \geq m.
    • Bornée (Bounded): A sequence is bornée if it is both minorée and majorée (m≤un≤Mm \leq u_n \leq M for every n∈In \in I).

  • Monotony (Direction of Variation):

    • Croissante (Increasing): For all n∈In \in I, un+1≥unu_{n+1} \geq u_n.
    • Décroissante (Decreasing): For all n∈In \in I, un+1≤unu_{n+1} \leq u_n.
    • Constante (Constant): For all n∈In \in I, un+1−un=0u_{n+1} - u_n = 0.

  • Methods to Study Monotony:

    • Method 1: Study the sign of (un+1−un)(u_{n+1} - u_n). If un+1−un≥0u_{n+1} - u_n \geq 0, the sequence is strictly increasing. If un+1−un≤0u_{n+1} - u_n \leq 0, the sequence is strictly decreasing.
    • Method 2: For sequences with strictly positive terms (un>0u_n > 0), compare the ratio un+1un\frac{u_{n+1}}{u_n} to 11.
      • If un+1un≥1\frac{u_{n+1}}{u_n} \geq 1, the sequence is increasing.
      • If un+1un≤1\frac{u_{n+1}}{u_n} \leq 1, the sequence is decreasing.
      • If terms are strictly negative, these conditions are reversed (un+1/un≥1  ⟹  u_{n+1} / u_n \geq 1 \implies decreasing).

Proof by Induction (Récurrence)

To prove a property PnP_n for all n≥n0n \geq n_0:

  1. Initialisation: Show that PnP_n is true for the starting rank n=n0n = n_0.
  2. Transmission (Heredity): Demonstrate that if PkP_k is true for an arbitrary integer k≥n0k \geq n_0 (Hypothesis of Induction), then Pk+1P_{k+1} is also true.
  3. Conclusion: State that PnP_n is true for all integers n≥n0n \geq n_0.

Convergence and Limits of Sequences

  • Monotone Convergence Theorem:

    • An increasing sequence that is majorée is convergent.
    • A decreasing sequence that is minorée is convergent.

  • Limits and Functions:

    • Given un=f(n)u_n = f(n): if lim⁡x→+∞f(x)=L\lim_{x \to +\infty} f(x) = L, then lim⁡n→+∞un=L\lim_{n \to +\infty} u_n = L.
    • Given un+1=f(un)u_{n+1} = f(u_n): if the sequence (un)(u_n) converges toward a limit LL and ff is continuous at LL, then LL is a solution to the equation f(L)=Lf(L) = L.

Practical Examples and Applications

  • Arithmetic Example: un=−3n+5u_n = -3n + 5.

    • un+1=−3(n+1)+5=−3n−3+5=−3n+2u_{n+1} = -3(n + 1) + 5 = -3n - 3 + 5 = -3n + 2.
    • un+1−un=(−3n+2)−(−3n+5)=−3u_{n+1} - u_n = (-3n + 2) - (-3n + 5) = -3.
    • The sequence is arithmetic with raison r=−3r = -3 and first term u0=5u_0 = 5.

  • Counter-Example (Non-Arithmetic): un=n2+1u_n = n^2 + 1.

    • u0=1,u1=2,u2=5u_0 = 1, u_1 = 2, u_2 = 5.
    • u1−u0=1u_1 - u_0 = 1, but u2−u1=3u_2 - u_1 = 3.
    • Since u1−u0≠u2−u1u_1 - u_0 \neq u_2 - u_1, the sequence is not arithmetic.

  • Geometric Example: un=25nu_n = \frac{2}{5^n}.

    • un+1=25n+1=25×5n=15×unu_{n+1} = \frac{2}{5^{n+1}} = \frac{2}{5 \times 5^n} = \frac{1}{5} \times u_n.
    • The sequence is geometric with raison q=15q = \frac{1}{5} and first term u0=2u_0 = 2.

  • Limit Calculations:

    • un=15nu_n = \frac{1}{5^n}: Since q=15∈]−1,1[q = \frac{1}{5} \in ]-1, 1[, lim⁡n→+∞un=0\lim_{n \to +\infty} u_n = 0.
    • vn=3×4nv_n = 3 \times 4^n: Since q=4>1q = 4 > 1 and v0=3>0v_0 = 3 > 0, lim⁡n→+∞vn=+∞\lim_{n \to +\infty} v_n = +\infty.
    • wn=−2×5nw_n = -2 \times 5^n: Since q=5>1q = 5 > 1 and w0=−2<0w_0 = -2 < 0, lim⁡n→+∞wn=−∞\lim_{n \to +\infty} w_n = -\infty.
    • en=(−3)ne_n = (-3)^n: Since q=−3≤−1q = -3 \leq -1, the sequence has no limit.

Logarithmic and Exponential Identities

  • Logarithmic Relations:
    • ln⁡(an)=n×ln⁡(a)\ln(a^n) = n \times \ln(a)
    • ln⁡(a×b)=ln⁡(a)+ln⁡(b)\ln(a \times b) = \ln(a) + \ln(b)
    • ln⁡(ab)=ln⁡(a)−ln⁡(b)\ln(\frac{a}{b}) = \ln(a) - \ln(b)
    • ln⁡(a)=12×ln⁡(a)\ln(\sqrt{a}) = \frac{1}{2} \times \ln(a)
  • Exponential Identities:
    • eln⁡(a)=ae^{\ln(a)} = a
    • ln⁡(er)=r\ln(e^r) = r where r∈Rr \in \mathbb{R}
  • Additional Property: If vn=ln⁡(un)v_n = \ln(u_n) and (un)(u_n) is a geometric sequence with raison qq, then (vn)(v_n) is an arithmetic sequence with raison r=ln⁡(q)r = \ln(q).

Common Pitfalls to Avoid

  • Do not assume that if a sequence is not arithmetic it must be geometric; a sequence can be neither.
  • Do not assume that if a sequence is defined by un+1=f(un)u_{n+1} = f(u_n), then the function ff and the sequence (un)(u_n) share the same variation direction.