Question 1: In an arithmetic sequence, the nth term is generally expressed as a formula: T(n) = a + (n - 1)d, where a is the first term and d is the common difference. This worksheet explores how to find this formula from given data and apply it to various problems.

Answer:

Question 1: Given the first term is 7 and the second term is 13, find the nth term formula for this sequence.

Answer: T(n) = 7 + (n - 1)6

Question 2: Sequence: 3, 8, 13, 18. Derive the nth term formula.

Answer: T(n) = 3 + (n - 1)5

Question 3: Find the first term and common difference if the 10th term is 37 and the 15th term is 42.

Answer: a = 2, d = 3

Question 1: A sequence has a first term of 5 and a common difference of 4. Find the 20th term. Then, explain how you can generalize the nth term formula from this information.

Answer: T(20) = 5 + (20 - 1)4 = 81. The nth term formula is T(n) = 5 + (n - 1)4.

Question 2: A sequence starts with 12, and the third term is 20. Find the general formula for the nth term and justify your steps.

Answer: Let a = 12, then find d using the third term: T(3) = 12 + 2d = 20, so 2d=8, d=4. Formula: T(n) = 12 + (n - 1)4.

Question 1: A company increases its savings by $50 each month. If the initial savings is $200, write an expression for the amount saved in month n. What will be the savings in month 12?

Answer: T(n) = 200 + (n - 1)50; T(12) = 200 + 11*50 = 750

Question 2: A cyclist completes 3 km on the first day and increases her distance by 2 km each subsequent day. Write the formula for distance traveled on day n and find her total distance after 10 days.

Answer: T(n) = 3 + (n - 1)2; Sum of first 10 terms: 10/2 * (T(1)+T(10)) = 5*(3 + 21) = 120 km.

Question 1: Given the 5th term is 20 and the 10th term is 35, find the nth term formula. Then, verify your formula by calculating the 7th term.

Answer: d = 3, a = 5; T(n) = 5 + (n - 1)3; T(7)= 5 + 6*3= 23

Question 2: A sequence has a first term of 9, and the 4th term is 21. The sequence's nth term formula is unknown. Derive the formula and explain your reasoning, then determine the 15th term.

Answer: T(4)= 9 + 3d= 21 => 3d=12 => d=4; T(n)= 9 + (n - 1)4; T(15)= 9 + 14*4= 65

Question 1: If T(1)= 4 and T(6)= 19, write the nth term formula and find T(10).

Answer: d=3, a=4; T(n)=4 + (n - 1)3; T(10)=4 + 9*3=31

Question 2: A sequence starts with 2 and has a common difference of 7. Write the formula and compute the 25th term.

Answer: T(n)=2 + (n - 1)7; T(25)=2 + 24*7=170

Question 1: A student claims the nth term is given as T(n) = a + n*d, missing the '(n - 1)' component. Explain why this is incorrect and show the correct formula.

Answer: The correct formula considers the first term at n=1: T(n)=a + (n - 1)d. Using T(1)=a, the adjustment ensures the formula matches the sequence's start.