Answer:
The equation of the nth term is an = -621 + 42n
Step-by-step explanation:
* Lets revise the arithmetic sequence
- There is a constant difference between each two consecutive numbers
- Ex:
# 2 , 5 , 8 , 11 , ……………………….
# 5 , 10 , 15 , 20 , …………………………
# 12 , 10 , 8 , 6 , ……………………………
* General term (nth term) of an Arithmetic sequence:
- U1 = a , U2 = a + d , U3 = a + 2d , U4 = a + 3d , U5 = a + 4d
- Un = a + (n – 1)d, where a is the first term , d is the difference
between each two consecutive terms
, n is the position of the term
* Lets solve the problem
∵ an = a + (n - 1)d
∴ a14 = a + (14 - 1)d
∴ a14 = a + 13d
∵ a14 = -33
∴ a + 13d = -33 ⇒ (1)
- Similar we can find another equation from a15
∵ a15 = a + (15 - 1)d
∴ a15 = a + 14d
∵ a15 = 9
∴ a + 14d = 9 ⇒ (2)
- We will solve equations (1) and (2) to find a and d
* Lets subtract equation (2) from equation (1)
∴ (a - a) + (13 - 14)d = (-33 - 9)
∴ -d = -42 ⇒ × both sides by -1
∴ d = 42
- Substitute this value of d in equation (1) or (2)
∵ a + 13d = -33
∵ d = 42
∴ a + 13(42) = -33
∴ a + 546 = -33 ⇒ subtract 546 from both sides
∴ a = -579
* Now lets write the equation of the nth term
∵ an = a + (n - 1)d
∵ a = -579 and d = 42
∴ an = -579 + (n - 1) 42 ⇒ open the bracket
∴ an = -579 + 42n - 42
∴ an = -621 + 42n
* The equation of the nth term is an = -621 + 42n
- its the result