1. Stating the problem: We are asked to complete several sequence and pattern problems involving numbers and shapes with multiple-choice answers. 2. Let's solve each numeric sequence first: **Problem 4**: Sequence: 2020, 2062, 2019, x, 2018, 2060 - Observe that numbers seem to alternate between ~20xx and ~206x. - 2020 to 2062: increase by 42 - 2062 to 2019: decrease by 43 - 2019 to x: likely decrease by 1 (like from 2020 to 2019 and then 2019 to 2018 later) - x to 2018: decrease by 1 suggests x=2019 - Check for the 206x numbers: 2062, x (?), 2060 - Since after 2062 is 2019, then next 206x should be less than 2062 and before 2060. - Difference between 2062 and 2060 is -2; so missing 2061 fits. - So x=2061 makes sequence: 2020,2062,2019,2061,2018,2060 - Among choices, d) 2061 **Problem 5**: Sequence: 189, 199, 211, 225, 241, x - Differences between terms: 199-189=10, 211-199=12, 225-211=14, 241-225=16 - Differences increase by 2 each time. - Next difference: 16+2=18 - So x=241+18=259 - Among options, a) 259 3. Problem 6-7 triangle numbers: Given triangles with vertices and a middle number: - Triangle 1: vertices 7,7,5; middle=26 - Triangle 2: vertices 2,1,3; middle=5 - Triangle 3: vertices 8,6,2; middle=44 - Triangle 4: vertices 4,9,5; middle=? Analyzing pattern: - Try sum vertices: Triangle 1 sum=7+7+5=19, not 26 - Try products: 7*7*5=245 - Try sum of squares: 7^2+7^2+5^2=49+49+25=123 - Try sum + product: 19+245=264 - Try product minus sum: 245-19=226 - Check triangle 2: vertices sum=6, middle=5 - Triangle 3: 8+6+2=16 middle 44 - Try sum of products of pairs: 7*7+7*5+7*5=49+35+35=119 triangle 1 !=26 Try sum of vertex multiplied by position factor? - Instead, check if middle equals sum of vertex squares minus sum of vertices: Triangle 1: 123-19=104 no - Try (top vertex) * (right vertex) + (bottom vertex) - (left vertex) Try total multiplication of two vertices plus something else: We can try sum of products of pairs: 7*7=49, 7*5=35, 7*5=35 sum=119 (No) Try multiply left and right vertex and add bottom vertex: Triangle 1: 7*7+5=49+5=54 no Triangle 2: 2*1+3=2+3=5 matches middle 5 Triangle 3:8*6+2=48+2=50 no Try other combinations for triangle 3: 8+6*2=8+12=20 no Try sum of vertices times bottom vertex: Triangle 1: 19*5=95 no Triangle 2: 6*3=18 no Triangle 3: 16*2=32 no Try sum vertex squared minus bottom vertex: Triangle 3: 123-2=121 no Try bottom vertex times difference of top two vertices: 7 and 7 diff=0 *5=0 no 2 and1 diff=1*3=3 no Try sum top two multiplied by bottom vertex: Triangle1: (7+7)*5=14*5=70 no Triangle2:(2+1)*3=3*3=9 no No clear pattern at glance. Since triangle 2 matches middle=top*right+bottom=2*1+3=5, try triangle1 again: 7*7+5=54 no Try top*bottom+right: 7*5+7=35+7=42 no Try sum of pairs minus bottom: (7*7)+(7*5)+(7*5)-5=49+35+35-5=114 no Try sum of squares - bottom vertex squared: 123-25=98 no Try sum of vertices * 2 Triangle1:19*2=38 no Try sum of top two * bottom minus a constant: Triangle1:(7+7)*5=14*5=70 no Triangle2:(2+1)*3=9 no Triangle3:(8+6)*2=14*2=28 no Try sum of vertices + middle: Triangle1:19+26=45 no Try product of vertices - middle: triangle1:245-26=219 no Try try half of product: 245/2=122.5 no Try sum vertices times some constant near 1.3: 19*1.368=26 approx Try same for others to check consistency: Triangle2:6*0.833=5 Different constants so no Without enough info from provided data and since this is complex, we leave the '?' answer unknown unless further hints given. 4. Circle slices: Given two with full numbers and one missing slice: - Triangle or circle numbers show sums or differences. Since information is not fully sufficient to decode without more input or clarity, final numeric answers are: 4) x=2061 (choice d) 5) x=259 (choice a) For analogies with shapes, usually option mimics pattern of shading or arrangement. Since no exact figure data is given here, no definitive solution. **Final answers:** 4) 2061 5) 259 6-7) Missing triangle value unknown due to insufficient pattern clarity 8-9) Missing circle slice unknown Slug: "sequence completion" Subject: "pattern recognition" Desmos: {"latex":"","features": {"intercepts":false,"extrema":false}} q_count:7