Sequences and Series Flashcards
What is an AP?
Arithmetic sequence/series, is an initial number (a) that is constantly added to by another number (d)
What is a GP?
Geometric sequence/series, is an initial number (a) that is constantly multiplied by another number (r)
What is a Tn?
Term number, the number seen in the nth term of an AP or GP sequence
Sequence → Term
What is an Sn?
Sum number, the sum of all the values of an AP or GP series up to the nth value
Series → Sum
Tn for an AP?
Tn = a + d(n-1)
Sn for an AP?
Sn = (n/2)(2a + d(n-1))
OR
Sn = (n/2)(a + l)
Tn for a GP?
Tn = ar^(n-1)
Sn for a GP?
Sn = a(1-r^n)/(1-r)
OR
Sn = a(r^n-1)/(r-1)
What is a limiting sum?
For a GP, some numbers will add up infinity to a finite value
Sl = a/(1-r)
given that -1 < r < 1 or |r| < 1
Proving by Induction
1. First step?
- Prove for the 1st term or show that for n = first term applies to both the first number, the last number and whatever follows the equal sign
E.g.
5 + 10 + 20 +…+ 5•2^n = 5(2^(n+1) - 1) [for n≥0]
5 should = 5•2(0) should = 5(2^(0+1) - 1)
5 = 5 = 5, correct ✔️
Proving by Induction
2. Second step?
- Assume n = k
E.g.
5 + 10 + 20 +…+ 5•2^n = 5(2^(n+1) - 1) [for n≥0]
assume n = k
5 + 10 + 20 +…+ 5•2^k = 5(2^(k+1) - 1) ✔️
Proving by Induction
3. Third step?
- Prove for n = k + following term
E.g.
5 + 10 + 20 +…+ 5•2^n = 5(2^(n+1) - 1) [for n≥0]
5 + 10 + 20 +…+ 5•2^(k) + 5•2(k + 1) = 5(2^(k+2) - 1)
5(2^(k+1) - 1) + 5•2(k + 1) = 5(2^(k+2) - 1)
5(2^(k+2) - 1) = 5(2^(k+2) - 1) ✔️
Proving by Induction
4. Fourth step?
- Write conclusion
Proving for n = (first term), assuming n = k and further proving for
n = k + (whatever value is necessary), we see it is true for the whole equation; proven by induction. (write something like this) ∴
