Sources: Visual - various maths sites (credits to original creator) Questions - Dong Zong's Textbook suitable for SUEC (Maths), SPM (Maths and Add Maths) too

1. 2

2. 𝒑𝒈 𝟏𝟏𝟑

3. 𝒑𝒈 𝟏𝟏𝟒

4. 𝒑𝒈 𝟏𝟏𝟒

8. 𝒑𝒈 𝟏𝟏𝟔

9. 𝒑𝒈 𝟏𝟏𝟔

10. 𝒑𝒈 𝟏𝟏𝟖

11. 𝒑𝒈 𝟏𝟏𝟔

12. 𝒑𝒈 𝟏𝟏𝟖

13. 𝒑𝒈 𝟏𝟏𝟕

14. 𝒑𝒈 𝟏𝟏𝟕
(𝟐) ÷ (𝟏)

15. 𝒑𝒈 𝟏𝟏𝟕

16. 𝒑𝒈 𝟏𝟏𝟕

17. 𝒑𝒈 𝟏𝟏𝟕
3 sides of triangle is AP.
perimeter = 12
area = 6
3 sides = ?
what kind of triange?
𝐿𝑒𝑡 3 𝑠𝑖𝑑𝑒𝑠 = 𝑎, 𝑎 + 𝑑, 𝑎 + 2𝑑
𝑎 + 𝑎 + 𝑑 + 𝑎 + 2𝑑 = 12
3(𝑎 + 𝑑) = 12
𝑎 + 𝑑 = 4
1
2
(𝑎)(𝑎 + 𝑑) = 6
𝑎 = 3
𝑑 = 1
∴ 3 𝑠𝑖𝑑𝑒𝑠 = 3, 4, 5
𝑎 𝑟𝑖𝑔ℎ𝑡 𝑎𝑛𝑔𝑙𝑒 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒
𝑎
𝑎 + 𝑑
𝑎 + 2𝑑

18. 𝒑𝒈 𝟏𝟏𝟗
sum of all multiples of 5 from 1 - 100

19. 𝒑𝒈 𝟏𝟐𝟏

20. 𝒑𝒈 𝟏𝟐𝟐

21. 𝒑𝒈 𝟏𝟐𝟐
sum of all multiples of 4 from 1 - 1000
𝑎 = 4, 𝑎𝑛 = 1000, 𝑑 = 4, 𝑛 =?
𝑎𝑛 = 4 + (𝑛 − 1) 4 = 1000
4 + 4𝑛 − 4 = 1000
𝑛 =
1000
4
𝑛 = 250
𝑆250 =
250
2
4 + 1000
= 125,500

22. 𝒑𝒈 𝟏𝟐𝟎
n = ? when Sn positive

23. 𝒑𝒈 𝟏𝟐𝟎

24. 𝒑𝒈 𝟏𝟐𝟐

25. 𝒑𝒈 𝟏𝟐𝟐
𝑇8 = 𝑎 + 7𝑑 = 0
𝑛 =? 𝑤ℎ𝑒𝑛 𝑆𝑛 = 0
𝑆𝑛 =
𝑛
2
2𝑎 + 𝑛 − 1 𝑑 = 0
𝑎 = −7𝑑 − −①
2𝑎 + 𝑛 − 1 𝑑 = 0 − −②
① 𝑖𝑛𝑡𝑜 ② 2(−7𝑑) + 𝑛 − 1 𝑑 = 0
𝑛𝑑 = 15𝑑
𝑛 = 15

26. 𝒑𝒈 𝟏𝟐𝟐
𝑇5 = 3, 𝑆10 = 26
1
4
, 𝑇𝑛 = 0 ?
𝑆10 =
𝑛
2
2𝑎 + 𝑛 − 1 𝑑 = 26
1
4
𝑇5 = 𝑎 + 4𝑑 = 3 − −①
5 2𝑎 + 9𝑑 = 26
1
4
8𝑎 + 36𝑑 = 21 − −②
𝑇𝑛 = 6 + (𝑛 − 1) −
3
4
= 0
𝑛 − 1 = 8
𝑛 = 9

27. 𝒑𝒈 𝟏𝟐𝟐

28. 𝒑𝒈 𝟏𝟐𝟐
𝑎 = 1, ∑𝐴𝑃 = 𝑜𝑑𝑑, 𝑆𝑜𝑑𝑑 = 175, 𝑆𝑒𝑣𝑒𝑛 = 150
𝑎𝑠𝑠𝑢𝑚𝑒 𝑆𝑜𝑑𝑑 = 1 + 3 + 5 + . . . + 2𝑛 − 1
𝑆𝑜𝑑𝑑 + 𝑆𝑒𝑣𝑒𝑛 =
13
2
2 + 12𝑑 = 175 + 150 = 325
𝑆𝑜𝑑𝑑 =
𝑛
2
(1 + 2𝑛 − 1) = 𝑛2
= 175
𝑛 ≈ 13
d = 4
𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒 =
𝑇13 = 1 + 12(4) = 49

29. 𝒑𝒈 𝟏𝟐𝟐
𝑎 = 182
− 172
𝑑 = 162 − 152 − (182 − 172)
= −4
𝑇𝑛 = 182
− 172
+ (𝑛 − 1)(− 4) = 22
− 12
n = 9
𝑆9 =
9
2
(182
− 172
+ 22
− 12
) = 171

30. 𝒑𝒈 𝟏𝟐𝟏

31. 𝒑𝒈 𝟏𝟐𝟏
𝑎 = 13
𝑆4 = 𝑆10

32. 𝒑𝒈 𝟏𝟐𝟐 − 𝟏𝟐𝟑

33. 𝒑𝒈 𝟏𝟐𝟐 − 𝟏𝟐𝟑
5 2+4+6+...+2𝑛
=
1
25
−28
5 2(𝟏+𝟐+𝟑+...+𝒏)
= 5−2 −28
5
2
𝒏
𝟐 𝟏+𝒏
= 556
𝑛(𝑛 + 1) = 56
𝑛2
+ 𝑛 − 56 = 0
(𝑛 − 7)(𝑛 + 8) = 0
𝑛 = 7 𝑜𝑟 𝑛 = −8 (𝑖𝑛𝑣𝑎𝑙𝑖𝑑)

34. 𝒑𝒈 𝟏𝟐𝟑
𝑠𝑢𝑚 𝑜𝑓 1 𝑡𝑜 2000 𝑡ℎ𝑎𝑡 𝑐𝑎𝑛𝑛𝑜𝑡 𝑏𝑒 𝑑𝑖𝑣𝑖𝑑𝑒𝑑 𝑏𝑦 5
𝑎 = 5, 𝑑 = 5, 𝑎𝑛 = 2000
𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒(𝑑𝑖𝑣𝑖𝑠𝑖𝑏𝑙𝑒 𝑏𝑦 5) = 5, 10, . . . , 2000
𝑆2000 = 1000(1 + 2000)
= 2,001,000
𝑇𝑛 = 5 + (𝑛 − 1)5 = 2000
𝑛 = 400
𝑆400 = 200(5 + 2000)
= 401,000
𝑆2000 − 𝑆400 = 2,001,000 − 401,000
= 1,600,000

35. 𝒑𝒈 𝟏𝟐𝟑
𝐺𝑖𝑣𝑒𝑛 𝑇9 = 2𝑇5 , 𝑓𝑖𝑛𝑑 𝑟𝑎𝑡𝑖𝑜 𝑜𝑓 𝑆9 ∶ 𝑆5
𝑇9 = 2𝑇5
a + 8d = 2(𝑎 + 4𝑑)
= 2𝑎 + 8𝑑
𝑆9
𝑆5
=
9
2
2𝑎 + 8𝑑
5
2
2𝑎 + 4𝑑
a = 2𝑎
a = 0
=
18
5

36. 𝑛 = 20 , 𝑆10 = 120 , 𝑆11 − 20 = 320
𝑓𝑖𝑛𝑑 𝑎, 𝑑, 𝑇15
𝑆20 = 𝑆10 + 𝑆21 − 20
= 120 + 320
= 440
𝑆20 = 10 2𝑎 + 19𝑑 = 440
𝑆10 = 5 2𝑎 + 9𝑑 = 120
2𝑎 + 9𝑑 = 24 − −①
2𝑎 + 19𝑑 = 44 − −②
𝑎 = 3, 𝑑 = 2
𝑇15 = 𝑎 + 14𝑑
= 31

37. 𝑎 = −23 , 𝑆𝑛 = −8 , 𝑇9 = 1 𝑓𝑖𝑛𝑑 𝑛, 𝑙
𝑆𝑛 =
𝑛
2
−46 + (𝑛 − 1) 3 = −8
𝑑 = 3
𝑇9 = 𝑎 + 8𝑑 = 1
−23 + 8𝑑 = 1
𝑛 (−46 + 3𝑛 − 3) = −16
3𝑛2
− 49𝑛 + 16 = 0
(3𝑛 − 1)(𝑛 − 16) = 0
𝑛 =
1
3
(𝑖𝑛𝑣𝑎𝑙𝑖𝑑) 𝑜𝑟 𝑛 = 16
𝑆16 = 8 −23 + 𝑙 = −8
𝑙 = 24