The number of dots given are 1, 4, 9, 11, 16, 25, ...
The no. of dots are in the pattern 1 \(\times\) 1, 2 \(\times\) 2, 3 \(\times\) 3, 4 \(\times\) 4, 5 \(\times\) 5 .... 10 \(\times\) 10
\(\therefore \) The number of dots in the 10^th figure = 100.

The single portions  A, B, C, D, E, F, G and H are 8 triangles. 
Taking the combination of 2 E & F, F & G, & H and E & H are 4 triangles.
There are 8 + 4 = 12 triangles.

6 possibilities,
i) R - B - R - B  
ii) R - R - B - B 
iii) B - R - R - B 
iv) B - R - B - R  
v) B - B - R - R  
vi) R - B - B - R
(R- RED, B- BLUE)

There are 12 straight lines used which is minimum.

(i) The next pattern is 
(ii)

Pattern	1	2	3	4	5
Number of dots	2	5	9	14	20

(iii)

Pattern	Number of dots
1	2
2	2+3
3	2+3+4
4	2+3+4+5

First number is 2
2^nd number is 2 + 3 = 5
3^rd number is 5 + 4 = 9
4^th number is 9 + 5 = 14
5^th number is 14 + 6 = 20
(iv) Number of dots in the 25^th pattern is

Pattern	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25
No. of dots	2	5	9	14	20	27	35	44	54	65	77	90	104	119	135	152	170	189	209	230	252	275	299	324	350

\(\therefore \) Number of dots in the 25^th pattern = 350.

(i) Yes
(ii) 5
(iii) 
     
Sums are 17, 19, 20, 21 and 23.

The number of dots given are 1, 3, 6, 10, 15, ...

Pattern	Number of dots
1	1
2	3
3	6
4	10
5	15

1^st number = 1
2^nd number = 1 + 2 = 3
3^rd number = 3 + 3 = 6
4^th number = 6 + 4 = 10
5^th number = 10 + 5 = 15
6^th number will be 15 + 6 = 21
7^th number will be 21 + 7 = 28
8^th number will be 28 + 8 = 36
9^th number will be 36 + 9 = 45
10^th number will be 45 + 10 = 55
Number of dots in the 10^th figure = 55.

Taking smaller triangles we have AFB, BFG, BGC, CGH, HCD, DHI, DIE, EIJ, EJA and AJF \(\Rightarrow \)10 triangles.
Taking the combination of 2.
CGD, CID, DJE, DHE, EIA, EFA, AGB, AJB, BHC and BFC are 10 triangles.
Combining 3 at a time we have CED, DAE, EBA, ACB, BDC, CEF, DBJ, GDA, EBH, ACI => 10 triangles.
Other triangles are BCE, CAD, DBE, ADB and ACE are 5 triangles.
\(\therefore \) Total 10 + 10 + 10 + 5 = 35 triangles.

(iii) Every part A, B, C, D, E, F, G, H, I, J, K, L, M, N, O and P are 16 triangles.
Combining two at a time. A & B, B & C, C & D, D & A, J & K, L & M, N & O, P & I are 8 triangles.
Taking the combination of 4, we have I, J, E & A; K, L, F and B, M, N, G and C, P, O, H and D are 4 triangles.
Taking 8 parts together we have QRS, RST, STQ, and TRQ are 4 triangles.
\(\therefore \) Total triangles = 16 + 8 + 4 + 4 = 32 triangles.