Flashcards · 13 cards

Sequences, series & binomial expansion

Say the answer to yourself, then check. Cards you know come back less and less often; cards you don't come back tomorrow.

Rule
The $n$ th term and the sum of $n$ terms of an A.P.?
Answer
$T_n = a + (n - 1)d$ and $S_n = \dfrac n2[2a + (n - 1)d]$ .
Sum of an A.P.Sₙ = n/2 [2a + (n − 1)d]
From the lesson: Harder arithmetic progressions
Know it
"How much in the 16th week?" and "how much in 16 weeks?": which is which?
Answer
The first is a term, $T_{16}$ . The second is a sum, $S_{16}$ .
From the lesson: Harder arithmetic progressions
Rule
The least number of terms for the sum to pass 90?
Answer
The first $n$ where $S_n$ goes past 90: $S_7 = 77 < 90 < 100 = S_8$ , so $n = 8$ . Solve the inequality, then round up.
First past the targetS₇ = 77 < 90 < 100 = S₈: least n is 8
From the lesson: Harder arithmetic progressions
Rule
$u_{n + 2} = u_{n + 1} + 2u_n$ , with $u_1 = u_2 = 1$ . Find $u_3$ .
Answer
Build each term from the ones before: $u_3 = u_2 + 2u_1 = 1 + 2 = 3$ .
Each term from the ones beforeu₃ = u₂ + 2u₁ = 1 + 2 = 3, and so on
From the lesson: Harder arithmetic progressions
Rule
The $n$ th term, the sum of $n$ terms and the sum to infinity of a G.P.?
Answer
$T_n = ar^{n - 1}$ , $S_n = \dfrac{a(r^n - 1)}{r - 1}$ , and $S_\infty = \dfrac{a}{1 - r}$ when $-1 < r < 1$ .
Sum to infinityS∞ = a ÷ (1 − r), when −1 < r < 1
From the lesson: Harder geometric progressions
Rule
Three consecutive terms of a G.P. with a known product: what do you call them?
Answer
$\dfrac ar$ , $a$ and $ar$ . Their product is $a^3$ , so $a$ comes out at once.
Three consecutive termsThe product is a³, so a comes out at once
From the lesson: Harder geometric progressions
Know it
$r^2 = 9$ . What is $r$ ?
Answer
$r = 3$ or $r = -3$ . Use the question's condition, such as " $r > 0$ ", to choose.
From the lesson: Harder geometric progressions
Rule
The coefficients of $(a + b)^5$ ?
Answer
Row 5 of Pascal's triangle: $1, 5, 10, 10, 5, 1$ , that is ${}^5C_0, \dots, {}^5C_5$ .
Pascal's triangleRow 5: 1, 5, 10, 10, 5, 1 are ⁵C₀ … ⁵C₅
From the lesson: The binomial expansion
Rule
The $(r + 1)$ th term of $(a + b)^n$ ?
Answer
${}^nC_r\, a^{n - r} b^r$ : the powers always add up to $n$ .
The (r + 1)th termⁿCᵣ aⁿ⁻ʳ bʳ: the powers always add up to n
From the lesson: The binomial expansion
Rule
$(1 + x)^n$ when $n$ is a fraction or negative?
Answer
$1 + nx + \dfrac{n(n - 1)}{2!}x^2 + \dfrac{n(n - 1)(n - 2)}{3!}x^3 + \dots$ , valid only for $|x| < 1$ . Write each coefficient out; don't use ${}^nC_r$ .
Close only for small x√(1 + x) and its first three terms agree inside |x| < 1 and pull apart outside it
From the lesson: The binomial expansion
Which method?
WAEC 2019 · Paper 2 · Q3
How many terms of the series $-3 - 1 + 1 + \ldots$ add up to 165?
How do you find how many terms add up to 165?
Answer
With $a = -3$ and $d = 2$ , $S_n = \frac n2[-6 + 2(n - 1)] = n(n - 4)$ . Set $n(n - 4) = 165$ and keep the positive whole-number solution.
From the lesson: Harder arithmetic progressions Try the question
Which method?
WAEC 2018 · Paper 2 · Q3
The sum of the second and third terms of a Geometric Progression (G.P.) is 48. If the sum of the third and fourth terms is 144, find the first term of the progression.
How do you find $r$ first?
Answer
Factorise each fact: $ar(1 + r) = 48$ and $ar^2(1 + r) = 144$ . Divide the second by the first: $r = 3$ .
From the lesson: Harder geometric progressions Try the question
Which method?
WAEC 2011 · Paper 2 · Q3
Write down the binomial expansion of $(2 - x)^5$ in ascending powers of $x$ .
Use your expansion in (a) to evaluate $(1.98)^5$ correct to four decimal places.
Which value of $x$ gives $1.98^5$ from the expansion of $(2 - x)^5$ ?
Answer
Solve $2 - x = 1.98$ : $x = 0.02$ . Put it into the expansion and round to the accuracy asked.
From the lesson: The binomial expansion Try the question