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Number and Algebra - Sequences and Series (Arithmetic and Geometric)

Grade 11IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Arithmetic Sequences: A sequence where each term increases or decreases by a constant value called the common difference dd. Visually, if you plot the term number nn on the horizontal axis and the term value unu_n on the vertical axis, the resulting points will lie on a straight line with a slope equal to dd.

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Geometric Sequences: A sequence where each term is found by multiplying the previous term by a constant ratio rr. On a graph, these sequences represent exponential change; if r>1r > 1, the points curve upwards rapidly (growth), whereas if 0<r<10 < r < 1, the points curve downwards, flattening towards the horizontal axis (decay).

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Sigma Notation: The symbol ∑\sum is used to represent the sum of a sequence. The index of summation (e.g., i=1i=1) is written below the symbol, the upper limit (last term number) is written above, and the general term formula is written to the right. It visually consolidates a long addition string into a compact mathematical expression.

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Common Difference and Ratio: The common difference in an arithmetic sequence is found by d=un+1−und = u_{n+1} - u_n. The common ratio in a geometric sequence is found by r=un+1unr = \frac{u_{n+1}}{u_n}. These values determine the 'step' size or 'scaling' factor between any two consecutive points in the sequence.

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Arithmetic Series: The sum of the terms of an arithmetic sequence. The sum SnS_n can be interpreted visually as the 'area' under the discrete steps of the sequence, often calculated by pairing the first and last terms to find an average value and multiplying by the number of terms nn.

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Geometric Series: The sum of the terms of a geometric sequence. Unlike arithmetic series which always diverge as nn increases, a geometric series can be finite or infinite. The total sum depends heavily on whether the multiplier rr causes the terms to expand or shrink.

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Convergence of Infinite Geometric Series: An infinite geometric series only has a finite sum if ∣r∣<1|r| < 1. Visually, this means that as nn approaches ∞\infty, the individual terms unu_n become so small they approach the x-axis, allowing the total sum to settle at a horizontal limit called the sum to infinity S∞S_{\infty}.

📐Formulae

un=u1+(n−1)du_n = u_1 + (n - 1)d

Sn=n2(u1+un)=n2(2u1+(n−1)d)S_n = \frac{n}{2}(u_1 + u_n) = \frac{n}{2}(2u_1 + (n - 1)d)

un=u1rn−1u_n = u_1 r^{n-1}

Sn=u1(rn−1)r−1=u1(1−rn)1−rS_n = \frac{u_1(r^n - 1)}{r - 1} = \frac{u_1(1 - r^n)}{1 - r}

S∞=u11−r, for ∣r∣<1S_{\infty} = \frac{u_1}{1 - r}, \text{ for } |r| < 1

💡Examples

Problem 1:

An arithmetic sequence has a first term u1=5u_1 = 5 and a common difference d=3d = 3. Find the 15th15^{th} term and the sum of the first 1515 terms.

Solution:

  1. To find the 15th15^{th} term, use the general term formula: u15=u1+(15−1)du_{15} = u_1 + (15 - 1)d.
  2. Substitute the values: u15=5+(14)(3)=5+42=47u_{15} = 5 + (14)(3) = 5 + 42 = 47.
  3. To find the sum of the first 1515 terms, use the sum formula: S15=152(u1+u15)S_{15} = \frac{15}{2}(u_1 + u_{15}).
  4. Substitute the values: S15=152(5+47)=7.5×52=390S_{15} = \frac{15}{2}(5 + 47) = 7.5 \times 52 = 390.

Explanation:

We first identify the sequence as arithmetic and use the linear growth formula for the specific term, then apply the summation formula which essentially averages the first and last terms and scales by the number of terms.

Problem 2:

A geometric sequence starts with 16,8,4,...16, 8, 4, .... Determine the common ratio rr and calculate the sum to infinity S∞S_{\infty}.

Solution:

  1. Find the common ratio rr by dividing the second term by the first term: r=816=0.5r = \frac{8}{16} = 0.5.
  2. Check if the sum to infinity exists: Since ∣0.5∣<1|0.5| < 1, the series converges.
  3. Use the infinite sum formula: S∞=u11−rS_{\infty} = \frac{u_1}{1 - r}.
  4. Substitute the values: S∞=161−0.5=160.5=32S_{\infty} = \frac{16}{1 - 0.5} = \frac{16}{0.5} = 32.

Explanation:

The sequence is geometric with a ratio less than 1, meaning the terms get smaller. This allows the series to converge to a finite 'limit' sum rather than growing indefinitely.