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On day 1 , Jane invites 5 friends to her party. She asks each of those friends to visit and invite 4 of their friends the next day, and to have each of their friends visit and invite 4 of their friends the day after, and so on. Let P be the number of people who are invited to Jane's party on day t. Assume that no person is invited more than once. Which of the following best explains the relationship between t and P ? Choose 1 answer: (A) The relationship is linear because all of the values of P are multiples of 10 . (B) The relationship is exponential because P increases by a factor of 4 each time t increases by 1 . (C) The relationship is exponential because P increases by a factor of 5 each time t increases by 1 . D The relationship is linear because P increases by a factor of 5 as t changes from 1 to 2 .

On day 11 , Jane invites 55 friends to her party. She asks each of those friends to visit and invite 44 of their friends the next day, and to have each of their friends visit and invite 44 of their friends the day after, and so on. Let P P be the number of people who are invited to Jane's party on day t t . Assume that no person is invited more than once. Which of the following best explains the relationship between t t and P P ?\newlineChoose 11 answer:\newline(A) The relationship is linear because all of the values of P P are multiples of 1010 .\newline(B) The relationship is exponential because P P increases by a factor of 44 each time t t increases by 11 .\newline(C) The relationship is exponential because P P increases by a factor of 55 each time t t increases by 11 .\newline(D) The relationship is linear because P P increases by a factor of 55 as t t changes from 11 to 22 .

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Q. On day 11 , Jane invites 55 friends to her party. She asks each of those friends to visit and invite 44 of their friends the next day, and to have each of their friends visit and invite 44 of their friends the day after, and so on. Let P P be the number of people who are invited to Jane's party on day t t . Assume that no person is invited more than once. Which of the following best explains the relationship between t t and P P ?\newlineChoose 11 answer:\newline(A) The relationship is linear because all of the values of P P are multiples of 1010 .\newline(B) The relationship is exponential because P P increases by a factor of 44 each time t t increases by 11 .\newline(C) The relationship is exponential because P P increases by a factor of 55 each time t t increases by 11 .\newline(D) The relationship is linear because P P increases by a factor of 55 as t t changes from 11 to 22 .
  1. Day 11 Invite: On day 11, Jane invites 55 friends. So, P=5P = 5 when t=1t = 1.
  2. Day 22 Invite: Each friend invites 44 more friends on the next day. So, on day 22, the number of new invites is 5×45 \times 4.
  3. Day 22 Total: On day 22, P=5+(5×4)P = 5 + (5 \times 4) because the original 55 friends are still counted.
  4. Day 33 New Invites: On day 33, each of the 2020 new people from day 22 invites 44 friends, so the number of new invites is 20×420 \times 4.
  5. Day 33 Total: On day 33, P=5+(5×4)+(20×4)P = 5 + (5 \times 4) + (20 \times 4). This pattern shows that with each day, the number of invites increases by a factor of 44 times the previous day's new invites.
  6. Relationship Type: The relationship between tt and PP is not linear because the number of new invites each day is not constant; it's multiplied by 44 each day.
  7. Exponential Growth: The relationship is exponential because the total number of people invited, PP, increases by a factor of 44 each time tt increases by 11.

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