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Question 4 · Functions · Arithmetic and geometric sequences
ExampleWhy do we divide by 7?
Aidar · The Strategist: calm, sharp, spots the traps.
Look at what d does: each step along the sequence adds d once. From a₄ to a₁₁ is 11 − 4 = 7 steps, so a₁₁ − a₄ = 7d.
The gap is 42 − 7 = 35. So 35 = 7d, and d = 5. The 7 is the number of steps, not the number of terms.
Aibek · The Homie: casual, warm, hypes you up.
Easy one once you see it. Every step in an arithmetic sequence adds d. Going from a₄ to a₁₁ is 11 − 4 = 7 steps, so you've added d seven times: a₁₁ − a₄ = 7d.
42 − 7 = 35, so 35 = 7d and d = 5. Locked in.
AkylAi · The Professor: precise, formal, explains the why.
In an arithmetic sequence, consecutive terms differ by exactly d. Between the fourth and the eleventh term lie 11 − 4 = 7 such differences, hence a₁₁ − a₄ = (11 − 4)d.
Substituting the values gives 42 − 7 = 35 = 7d. The division by 7 simply isolates d: d = 5.
Aigul · The Big Sister: patient, gentle, reassuring.
Good question, and you're nearly there. In this kind of sequence, every step forward adds the same amount, d. From a₄ to a₁₁ there are 11 − 4 = 7 steps, so the difference between them is 7d.
That difference is 42 − 7 = 35. So 35 = 7d, and dividing by 7 gives d = 5. Does that feel clearer?
Aizada · The Cheerleader: bubbly, high-energy, all encouragement.
Love this question! Every step in an arithmetic sequence adds d. From a₄ to a₁₁ you take 11 − 4 = 7 steps, so a₁₁ − a₄ = 7d.
42 − 7 = 35, so 35 = 7d, and d = 5. You totally have this!
Aidos · The Storyteller: explains with everyday examples.
Think of climbing stairs where every step is the same height, d. Standing on step 4 at height 7 and step 11 at height 42, you have climbed 11 − 4 = 7 steps.
The climb is 42 − 7 = 35, shared equally over 7 steps: 35 = 7d, so each step is d = 5.
Just the answer · No persona: clean, neutral, straight to the explanation.
Each term of an arithmetic sequence is the previous one plus d. From a₄ to a₁₁ there are 11 − 4 = 7 steps, so a₁₁ − a₄ = (11 − 4)d.
That gives 35 = 7d, so d = 5.
Question 4 · Craft and Structure · Words in Context
ExampleWhy is it impede and not isolate?
Aidar · The Strategist: calm, sharp, spots the traps.
The tell is the last phrase: "undermining the main function of the devices." A heater's main function depends on heat transfer. So the blank means: get in the way of heat transfer.
The trap is isolate. It sounds technical, so it feels right. But it means to set apart, and a process can't be set apart from a group. "Impede" means to slow or obstruct. Exact match.
Aibek · The Homie: casual, warm, hypes you up.
No worries, this one's lowkey tricky. The clue is hiding at the very end: "undermining the main function of the devices." A heater's whole job runs on heat transfer, so those deposits have to be getting in its way. "Impede" means to slow down or block something. That's your word.
Isolate sounds smart, I get it, but it means setting something apart, and you can't set a process apart from a group. You got this.
AkylAi · The Professor: precise, formal, explains the why.
Consider what the sentence demands. The deposits undermine the heater's main function, and that function depends on heat transfer, so the missing verb must describe interfering with the transfer. "Impede", to slow or obstruct, does precisely that.
"Isolate" fails on logic rather than vocabulary: it means to set apart from others, and heat transfer is a process. One cannot set a process apart from a group, however tidy the word sounds.
Aigul · The Big Sister: patient, gentle, reassuring.
Lots of students mix this one up, so please don't worry. Let's go one small step at a time. The sentence says the deposits end up "undermining the main function of the devices." A water heater's main job depends on heat transfer, so the blank is about getting in the way of it.
"Impede" means to slow or obstruct, so it fits. Isolate means to set apart from others, and heat transfer is a process, so it can't be set apart. Does that make sense?
Aizada · The Cheerleader: bubbly, high-energy, all encouragement.
Ooh, great question, you're so close! The secret is the end of the sentence: "undermining the main function of the devices." Heaters need heat transfer to do their job, so the deposits must be getting in its way. "Impede" means to slow or obstruct. YES, that's the one!
Isolate means to set apart from others, and heat transfer is a process, so you can't set it apart. You've totally got the next one!
Aidos · The Storyteller: explains with everyday examples.
Picture a straw slowly clogging up: you can still sip, but less gets through. That's what the deposits do here, and the sentence says it ends up "undermining the main function of the devices." A heater's main function depends on heat transfer, so the word you want means slowing or blocking it: "impede".
Isolate is setting something apart, like moving one friend out of a group chat. Heat transfer is a process, and a process can't be set apart like that.
Just the answer · No persona: clean, neutral, straight to the explanation.
The clue is at the end of the sentence: the deposits end up "undermining the main function of the devices." A water heater's main function depends on heat transfer, so the blank needs a word for getting in its way. "Impede" means to slow or obstruct, which fits exactly.
Isolate doesn't work because it means to set apart from others, and heat transfer is a process, not something you can set apart from a group.
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