Flashcards · 14 cards

Sequences & series (AP, GP)

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Rule
The $n$ th term of an A.P.?
Answer
$T_n = a + (n - 1)d$ , where $a$ is the first term and $d$ the common difference.
nth term of an A.P.Tₙ = a + (n − 1)d: T₅ has 4 blocks of d
From the lesson: Sequences and arithmetic progressions
Know it
What is the common difference of $20, 17, 14, \dots$ ?
Answer
Second term minus first: $17 - 20 = -3$ .
From the lesson: Sequences and arithmetic progressions
Rule
$x$ , $y$ and $z$ are consecutive terms of an A.P. What is true?
Answer
$y - x = z - y$ , so $2y = x + z$ .
Consecutive terms of an A.P.y − x = z − y, so 2y = x + z
From the lesson: Sequences and arithmetic progressions
Know it
How do you check a formula for the $n$ th term?
Answer
Test it on at least three terms. Many formulas fit the first two.
From the lesson: Sequences and arithmetic progressions
Rule
The sum of the first $n$ terms of an A.P.?
Answer
$S_n = \dfrac n2(a + l)$ when you know the last term $l$ , or $S_n = \dfrac n2\big[2a + (n - 1)d\big]$ .
Sum of an A.P.Two copies make n columns of a + l, so Sₙ = n/2 (a + l)
From the lesson: The sum of an arithmetic progression
Know it
How many terms are there in $2 + 5 + 8 + \dots + 62$ ?
Answer
Use the $n$ th term, not the last number: $2 + (n - 1) \times 3 = 62$ , so $n = 21$ .
From the lesson: The sum of an arithmetic progression
Rule
The $n$ th term of a G.P.?
Answer
$T_n = ar^{n - 1}$ : only $r$ is raised to the power. For $3, 6, 12, \dots$ , $T_6 = 3 \times 2^5 = 96$ .
nth term of a G.P.Tₙ = arⁿ⁻¹: each bar is r times the one before
From the lesson: Geometric progressions
Rule
$x$ , $y$ and $z$ are consecutive terms of a G.P. What is true?
Answer
$\dfrac yx = \dfrac zy$ , so $y^2 = xz$ .
Consecutive terms of a G.P.y ÷ x = z ÷ y, so y² = xz
From the lesson: Geometric progressions
Know it
$r^2 = 4$ . What is $r$ ?
Answer
$r = 2$ or $r = -2$ . An even power gives two values; $r^3 = 8$ gives only $r = 2$ .
From the lesson: Geometric progressions
Rule
The sum of the first $n$ terms of a G.P.?
Answer
$S_n = \dfrac{a(r^n - 1)}{r - 1}$ .
Sum of a G.P.rS − S = arⁿ − a, so Sₙ = a(rⁿ − 1) ÷ (r − 1)
From the lesson: The sum of a G.P. and the sum to infinity
Rule
The sum to infinity of a G.P.? When does it exist?
Answer
$S_\infty = \dfrac{a}{1 - r}$ , only when $-1 < r < 1$ .
Sum to infinityWith r = ½ each piece is half the gap left: S∞ = a ÷ (1 − r)
From the lesson: The sum of a G.P. and the sum to infinity
Which method?
WAEC 2022 · Paper 2 · Q1
The third term of an Arithmetic Progression (A.P.) is 23 and the sum of the first seven terms is 210. Find the:
common difference;
first term;
sum of the first 20 terms.
How do you turn each fact into an equation?
Answer
The third term is one term: $a + 2d = 23$ . The sum of seven terms is a sum: $\frac72[2a + 6d] = 210$ . Solve the two together.
From the lesson: The sum of an arithmetic progression Try the question
Which method?
WAEC 2024 · Paper 2 · Q6
Given that $(x + 2)$ , $(4x + 3)$ and $(7x + 24)$ are consecutive terms of a geometric progression (G.P.), find the:
values of $x$ ;
common ratio (for each value of $x$ ).
What equation links the three terms?
Answer
In a G.P. the ratios are equal: $\dfrac{4x + 3}{x + 2} = \dfrac{7x + 24}{4x + 3}$ . Cross-multiply to get a quadratic.
From the lesson: Geometric progressions Try the question
Which method?
JAMB 1999 · UME · Q12
The first term of a geometric progression is twice its common ratio. Find the sum of the first two terms of the progression if its sum to infinity is 8.
How do you use "the first term is twice its common ratio"?
Answer
Write $a = 2r$ and put it into $S_\infty = \dfrac{a}{1 - r} = 8$ . That leaves one unknown, $r$ .
From the lesson: The sum of a G.P. and the sum to infinity Try the question