Mathematics 10

1. P A T T E R

2. 1. What is the next shape?
, , , , , , ,
____________
2. What is the next number?
0, 4, 8, 12, 16, _______
3. What is the 7th number?
160, 80, 40, 20, 10, …?

3. Is a function whose domain is either
finite set or infinite set arranged in
order.

5. A= ( 5, 10, 15, 20, 30, 40)
B= (5, 10, 15, 20, 25, 30)

6. 1.𝑎𝑛 = 𝑛 + 4
2. 𝑎𝑛 = (−2)𝑛

7. Count the number of matchsticks in each figure and record
results on the table.
Number of
squares
1 2 3 4 5 6 7 8 9 1
0
Number of

8. A sequence where every term after
the first is obtained by adding a
constant called the common
difference

9. 1.17, 14, ___, ___, 5
2.13, ___, ___, ___, -11, -17

10. 𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑

11. Finding the 10th term:
𝑎10=13+(10−1)−6
𝑎10= 13+ (9)-6
𝑎10= 13+ (9)-6
𝑎10= 13-54
𝑎10= -41

12. 1. Find the 13th term of the sequence 5, 8, 11,
…
2. Find the 6th term of the sequence 6, 3, 0, …
Name 𝑎1 𝑎𝑛𝑑 𝑑 first before solving. (3 points
each)

13. The terms between any two non-
consecutive terms of an arithmetic
sequence.

14. Insert 4 arithmetic means between 5 and 25.
𝑎1 = 5 𝑎6 = 25
Using the formula for arithmetic sequence,
𝑎6 = 𝑎1 + 6 − 1 𝑑
𝑎6 = 𝑎1 + 5 𝑑
25= 5 + 5𝑑
25-5= 5d
20= 5d
d=4

15. 𝑎2 = 𝑎1 + 2 − 1 𝑑
𝑎2 = 5 + 1 4
𝑎2 = 9
(5, 9, 13, 17, 21, 25)

16. Find 3 terms between 2 and 34 of an arithmetic
sequence.
a. Solve for d.
b. Solve for 𝑎2
c. Solve for 𝑎3
d. Solve for 𝑎4
e. What is the arithmetic sequence?

17. 𝑆𝑛 =
𝑛
2
(𝑎1 + 𝑎𝑛)
𝑆𝑛 =
𝑛
2
[𝑎1 + (𝑎1+ 𝑛 − 1 ]𝑑
𝑆𝑛 =
𝑛
2
[2𝑎1 + 𝑛 − 1 𝑑]

18. Find the sum of the first 10 terms of arithmetic
sequence 5, 9, 13, 17, …
To solve d,
d= 9 - 5= 4
Using the formula,
𝑆𝑛 =
𝑛
2
[2𝑎1 + 𝑛 − 1 𝑑]
𝑆𝑛 =
10
2
[2(5) + 10 − 1 4]
𝑆𝑛 = 5[10 + 9(4)]
𝑆𝑛 = 5 46 = 230

19. 1. Find the sum of the first 20 terms of
the arithmetic sequence -2, -5, -8, -11,
…
2. Find the sum of the first 7 terms of the
arithmetic sequence 6, 2, -2, -6, …

20. 1. The first term of an arithmetic sequence is 4 and the tenth term is
67.
What is the common difference? (1 pt.)
2. What is the thirty-second term of the arithmetic sequence -12, -7, -
2, 3, ... ? (2 pts.)
a. Find d b. Find 𝑎32
3. What is the sum of the first sixteen terms of the arithmetic
sequence
1, 5, 9, 13, ... ? (2 pts.)
a. Find d b. Find 𝑆16
4. What is the sum of the eleventh to twentieth terms (inclusive)of
the arithmetic sequence 7, 12, 17, 22, ... ? (Solve for d first before
finding the sum of 11th to 20th terms for 5 pts.)

21. 1. 2, 8
2. -3, 9
3. 1, ½
4. -5, -10
5. 12, 4

22. ARITHMETIC
SEQUENCE
5, 20, 80, 320, …

23. is a sequence where each term after the
first is obtained by multiplying the
preceding term by a non-zero constant
called common ratio.
Common ratio, r, can be determined by
dividing any term in a sequence that
precedes it.

24. 1. 5, 15, 45, 135, …
2. 7 2, 5 2, 3 2, 2, …
3.
1
3
,
2
3
,
3
3
,
4
3
, …
4. 5, −10, 20, −40
5. 8, 4, 2, 1, …

25. In arithmetic sequence, the
constant is the common
difference while in geometric
sequence, the constant is the
common ratio.

26. 1.3, 12, 48, ___, ___
2.¼, ___, ___, ___, 64, 256

27. an= a1 rn-1

28. What is the 10th term in the
geometric sequence 8, 4, 2, 1, …?

29. Find the 6th term of the geometric
sequence whose second term is 6 and
common ratio is 2.

30. Geometric means are the terms
between any two given terms in a
geometric sequence.

31. Insert three terms between 2 and
32 of a geometric sequence.

32. Insert two terms between 5 and
135 of a geometric sequence.

33. The geometric mean between the
first two terms in a geometric
sequence is 32. If the 3rd term is 4,
find the first term.

34. I. Fill in the blanks with the correct word found on
the box.
A geometric sequence contains term in which after
the first is obtained by __________ the preceding term
by a non-zero _____________ called common ratio. The
terms between any two terms in geometric sequence
is called ____________________. In every solution to a
geometric sequence, ______________ is the first thing
to be considered. In every geometric sequence,
___________________ serves to be the constant value.

35. II. Solve the following problems:
1. In the geometric sequence 6, 12, 24, 48, …, which term is 768?
a. Solve for r
b. Solve for n
2. Find k so that the terms k-3, k+1 and 4k-2 form a geometric sequence.
a. Solve for k
b. Solve for k-3, k+1 and 4k-2 (3pts)
c. Form a geometric sequence consisting of 5 terms
3. Find two geometric means between 2xy and 16xy4
a. Find r
b. Find the two geometric means (2 pts.)

36. Consider the geometric sequence 3, 6, 12, 24, 48, 96,
…, what is the sum of the first 5 terms?
a. Manually solve for the sum of the first 5 terms
b. Consider other solution below:
𝑆5 = 3 + 6 + 12 + 24 + 48
− 2𝑆5 = 6 + 12 + 24 + 48 + 96
−𝑆5= 3 − 96
𝑆5 = 93
How about if it will be applied in the formula?

37. 𝑆𝑛 = 𝑎1 + 𝑎1𝑟 + 𝑎1𝑟2 + 𝑎1𝑟3 + ⋯ + 𝑎1𝑟𝑛−1
−(r𝑆𝑛 = 𝑎1𝑟 + 𝑎1𝑟2 + 𝑎1𝑟3 + ⋯ + 𝑎1𝑟𝑛−1 + 𝑎1𝑟𝑛)
𝑆𝑛 − 𝑟𝑆𝑛 = 𝑎1 − 𝑎1𝑟𝑛
by factoring: 𝑆𝑛(1 − 𝑟) = 𝑎1(1−𝑟𝑛
)
1 − 𝑟
To get the sum:𝑆𝑛 =
𝑎1(1−𝑟𝑛)
1−𝑟

38. Find the sum of the first 4 terms of
the sequence 81, 27, 9, 3, 1, …

39. 𝑆𝑛 = 𝑎1(1−𝑟𝑛
)
1 − 𝑟
Will be used of r≠1.
What if r=1?
𝑆𝑛 = 𝑛𝑎1

40. -2, 2, -2, 2, -2, …
1. Find the sum of the first 5 terms
2. Find the sum of the first 8 terms
3. Find the sum of the first 4 terms
4. Find the sum of the first 3 terms
5. Find the sum of the first 7 terms

41. If r= -1, the sum Sn
simplifies to,
Sn= 0 if n is even
Sn= a1 if n is odd

42. 1.5 terms of 4, 12, 36, 108, … (2
pts)
2.6 terms of -3, 3, -3, 3, …
3.7 terms of 9, 9, 9, 9, 9, …
4.9 terms of 4, -4, 4, -4, ...