Free Printable Week 6: Sequences, Series, Binomial Theorem and Induction

1.

Write the explicit formula for the geometric sequence.

3, -12, 48, -192, ...

a)

a_n = 4(3)^n-1

b)

a_n = 3(4)^n-1

c)

a_n = -4(3)^n-1

d)

a_n = 3(-4)^n-1

2.

Find the common ratio (multiplier): a₁ = 4 and a₄ = 256

a)

84

b)

64

c)

4

d)

8

3.

What is the 14th term of the geometric sequence?

3, 9, 27, 81, ...

a)

14,348,907

b)

4,782,969

c)

1,594,323

d)

45

4.

Find the partial sum $S_6$ for the geometric series.

$-\frac{4}{5}+8-80+...$

a)

$S_6$ = 21,845

b)

S_6

c)

S_6

d)

S_6

5.

Which of the series below is CONVERGENT?

a)

$a_1=4$
$a_n=a_{n-1}\cdot\left(1.5\right)$

b)

$a_n=-n^2-8n+106$

c)

$a_n=\left(\frac{5}{10}\right)^n$

d)

$a_n=5n+6$

6.

$\sum_{n=4}^{\infty}2\left(0.5\right)^{n-1}$

(a)

7.

$\sum_{k=1}^{\infty}\left(\frac{2}{5}\right)^k$ Find the sum

a)

$\frac{2}{5}$

b)

$\frac{2}{3}$

c)

0

d)

$\infty$

8.

What is row 5 of Pascal's Triangle?

a)

1, 2, 1

b)

1, 3, 3, 1

c)

1, 4, 9, 4, 1

d)

1, 5, 10, 10, 5, 1

9.

Expand $\left(2x+4y\right)^3$

a)

$8x^3+48x^2y+96xy^2+64y^3$

b)

$40x^3+160x^2y+320xy^2+320y^3$

c)

$32x^3+48x^2y+96xy^2+64y^3$

d)

$8x^3+16x^2y+32xy^2+64y^3$

10.

Calculate the coefficient of $a^4b^6$ in the expansion of $\left(a+b\right)^{10}$ .

(a)

11.

Find the binomial coefficient

(a)

12.

Consider the expansion of $\left(x+b\right)^{30}$ .
What is the 4th term's coefficient?

(a)

13.

Which of these is the first step in mathematical induction?

a)

Prove the statement is true for the first element in the set.

b)

Show that if the statement is true for the first k elements, then it is true for the (k+1)st case.

c)

Prove that the problem you are working on is the base to all proofs.

d)

None of these are correct.

14.

Which of the following is the induction step in mathematical induction?

a)

Show that if the statement is true for the first k elements, then it is true for the (k+1)st element in the set.

b)

Show that the statement is true for the first few elements in the set.

c)

Show that your math problem is different from all other math problems.

d)

None of these are correct.

15.

When using mathematical induction to prove : $\sum_{i=1}^ni^2=\frac{n\left(n+1\right)\left(2n+1\right)}{6}$ . In step #2, after you have made your assumption, what are you trying to prove? (What is your goal?)

a)

$\sum_{i=1}^{k+1}i^2=\frac{\left(k\right)\left(k+1\right)\left(2k+1\right)}{6}+\left(k+1\right)^2$

b)

$S_{k+1}=\frac{k\left(k+1\right)\left(2k+1\right)}{6}$

c)

$\sum_{k=1}^n\left(k+1\right)^2=\frac{\left(k+1\right)\left(k+2\right)\left(2k+3\right)}{6}$

d)

$S_{k+1}=\left(k+1\right)^2$

Week 6: Sequences, Series, Binomial Theorem and Induction

Total questions: 15

Worksheet time: 4hrs 45mins

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