Near a distant planet, Emek and Tara prepared to deploy communication probes in six stages. The mission timeline showed that each stage would release 4 more probes than the previous stage, beginning with 7 probes in Stage 1. Tara recognized an arithmetic sequence with first term 7 and common difference 4, so she modeled the number released at Stage n as a_n = 7 + 4(n - 1). For Stage 6, the rule gave 27 probes. Emek pointed out that this was only the number released during that stage, not the total deployed by then. To find the total, they used the arithmetic series formula S_n = n(a_1 + a_n)/2. For six stages, S_6 = 6(7 + 27)/2 = 102 probes. The spacecraft carried 110 probes, so completing the plan would leave 8 unused. By distinguishing the sixth term from the sum of the first six terms, Emek and Tara confirmed that the expanding network could be completed without exceeding their supply.

Challenges

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Challenge 1 of 5

Challenges

Start
Challenge 1 of 4
Hint
Distinguish between a term of a sequence and the sum of several terms.
A student says that because a_6 = 27, only 27 probes have been deployed by the end of Stage 6. Which statement best explains the error?
The value 27 is the number released in Stage 6 only; the total deployed requires adding the releases from Stages 1 through 6.The value 27 is the total for the first 6 stages, but 6 more probes must be added for Stage 6.The value 27 is incorrect because an arithmetic sequence must have a common ratio.The value 27 represents the number of unused probes after Stage 6.
Correct. The sixth term describes Stage 6 alone, while the series sum gives the total through Stage 6.
Not quite. Consider what a_6 represents compared with S_6.
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Hint
Stage 6 uses 27 probes, and each new stage uses 4 more than the previous stage.
If the spacecraft tried to add Stage 7 while continuing the same pattern of releasing 4 more probes each stage, how many probes would be required for Stage 7?
Correct. Stage 7 would require 31 probes.
Check the common difference and apply it once to the Stage 6 amount.
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Hint
Compare the number of probes remaining after Stage 6 with the number required specifically for Stage 7.
The spacecraft carries 110 probes, and the first six stages use 102 probes. If Emek and Tara consider adding Stage 7 while keeping the same arithmetic pattern, which conclusion is correct?
They can complete Stage 7 because 110 - 102 = 8, and Stage 7 requires only 4 additional probes.They can complete Stage 7 because the seventh term is 31, which is less than 110.They cannot complete Stage 7 because it requires 31 probes, but only 8 probes remain after Stage 6.They cannot complete Stage 7 because the total after six stages is already greater than 110.
Correct. Only 8 probes remain, which is not enough for the 31 probes required in Stage 7.
Not quite. Find the remaining supply after six stages and compare it with the full Stage 7 requirement.
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Genişleyen Sonda Ağı
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ARC_G12_PA_02_STORY_03
Genişleyen Sonda Ağı
The Expanding Probe Network
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ARC_G12_PA_02_STORY_03
The Expanding Probe Network