Grade 11 · Quadratic sequences · 177 of 600
Quadratic sequences · Quadratic coefficient a=-1.5; Linear coefficient b=-1.5
Use index n=6. Calculate u₆, the forward first difference u₇−u₆, and the second difference u₈−2u₇+u₆. Givens: Quadratic coefficient a=-1.5; Linear coefficient b=-1.5.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
The sequence is uₙ=(-1.5)n²+(-1.5)n for integer n≥0. Use index n=6. Calculate u₆, the forward first difference u₇−u₆, and the second difference u₈−2u₇+u₆.
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A starting point
Substitute n=6,7,8 directly or simplify Δuₙ=a(2n+1)+b and Δ²uₙ=2a.
Work through the reasoning
Step 1
Identify the model and target
The sequence is uₙ=(-1.5)n²+(-1.5)n for integer n≥0. Use index n=6. The governing relation is uₙ=an²+bn; Δuₙ=a(2n+1)+b; Δ²uₙ=2a. Substitute n=6,7,8 directly or simplify Δuₙ=a(2n+1)+b and Δ²uₙ=2a.
Step 2
Substitute and calculate
At n=6, uₙ=(-1.5)(6)²+(-1.5)(6)=-63. The next difference is (-1.5)(2(6)+1)+(-1.5)=-21, and its next change is 2(-1.5)=-3.
Step 3
Check the mathematical meaning
Direct values are u₆=-63, u₇=-84, u₈=-108. Their differences reproduce -21 and -3. Index n: 6; Term value: -63; Next first difference: -21; Second difference: -3. Decimal values are rounded, so use unrounded intermediate values.
The answer
At n=6, uₙ=(-1.5)(6)²+(-1.5)(6)=-63. The next difference is (-1.5)(2(6)+1)+(-1.5)=-21, and its next change is 2(-1.5)=-3. Direct values are u₆=-63, u₇=-84, u₈=-108. Their differences reproduce -21 and -3. Animation check: Index n: 6; Term value: -63; Next first difference: -21; Second difference: -3. Decimal displays are rounded; retain the original parameters and exact π until the final step.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
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Watch the relationship
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The mathematical idea
A quadratic sequence has changing first differences but constant second differences. The linear coefficient changes every first difference by the same amount, while the quadratic coefficient sets the second difference. The case a=0 becomes a linear sequence rather than a quadratic one. The sequence is uₙ=(-1.5)n²+(-1.5)n for integer n≥0. Use index n=6. The requested state occurs at 60% playback; use the exact target stated in the question for your calculation.
uₙ=an²+bn; Δuₙ=a(2n+1)+b; Δ²uₙ=2a
03 · Reflect and transfer
Explain what changes and why.
Explain why the linear term changes the first differences but cancels from every second difference.
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.