Answer:
The explicit rule of the given sequence is a[tex]_{n}[/tex] = -84 + 7n
Step-by-step explanation:
The nth term of the arithmetic sequence is a[tex]_{n}[/tex] = a + (n - 1)d, where
- d is the common difference between each 2 consecutive terms
∵ a[tex]_{11}[/tex] = -7
∴ n = 11
→ Substitute n by 11 in the nth term rule
∴ a[tex]_{11}[/tex] = a + 10d
→ Equate the right sides of a[tex]_{11}[/tex]
∴ a + 10d = -7 ⇒ (1)
∵ a[tex]_{19}[/tex] = 49
∴ n = 19
→ Substitute n by 19 in the nth term rule
∴ a[tex]_{19}[/tex] = a + 18d
→ Equate the right sides of a[tex]_{19}[/tex]
∴ a + 18d = 49 ⇒ (2)
→ Subtract equation (1) from equation (2)
∵ (a - a) + (18d - 10d) = (49 - -7)
∴ 0 + 8d = (49 + 7)
∴ 8d = 56
→ Divide both sides by 8 to find d
∴ d = 7
→ Substitute the value of d in equation (1) to find a
∵ a + 10(7) = -7
∴ a + 70 = -7
→ Subtract 70 from both sides
∵ a + 70 - 70 = -7 - 70
∴ a = -77
→ Substitute the values of a and d in the rule of the nth term above
∵ a[tex]_{n}[/tex] = -77 + (n - 1)(7)
∴ a[tex]_{n}[/tex] = -77 + (n)(7) - (1)(7)
∴ a[tex]_{n}[/tex] = -77 + 7n - 7
∴ a[tex]_{n}[/tex] = -84 + 7n
∴ The explicit rule of the given sequence is a[tex]_{n}[/tex] = -84 + 7n
