Bellwork A) Write the explicit expression (equation) for the following arithmetic sequence, then find the 10 th term: 6, 3, 0, -3, … B) Write the explicit expression (equation) for the following geometric sequence, then find the 10 th term: 1, 5, 25, 125, …

A) Write the explicit expression (equation) for the following arithmetic sequence, then find the 10 th term: 6, 3, 0, -3, …

B) Write the explicit expression (equation) for the following geometric sequence, then find the 10 th term: 1, 5, 25, 125, …

End in Mind: Work with a partner to complete  You are filling a 200 gallon tank of water with a large hose. After some time passes the gauge shows that there is now 35 gallons in the tank. You're not sure how long ago you started filling it, so you leave and come back exactly 15 minutes later and see that it is now filled to 105 gallons. How long do you have until the tank begins to overflow?

Arithmetic: Determine the explicit expression given the common difference and a term What do we need to know?What do we already know? The 5 th term is 19. The common difference is +6. What is the first term? Plug in what you know to figure out what you don’t!

Arithmetic: Determine the explicit expression given the common difference and a term 19=a 1 +(5-1)6 a 5 =19d=6

Arithmetic: Determine the explicit expression given the common difference and a term 19=a 1 +(5-1)6 Now Simplify, and solve for a 1 19=a 1 +(4)6 19=a =a 1 Plug in what you know to write the explicit expression a n =-5+(n-1)6 Remember- This was given to begin with!

Arithmetic: Determine the explicit expression given the common difference and a term a n =-5+(n-1)6 Now we just need to simplify! a n =-5+6n-6 a n =6n-11

Arithmetic: Determine the explicit expression given the 1 st term and another term What do we need to know?What do we already know? The 1 st term is -8. The 6 th term is 22. What is the common difference? Plug in what you know to figure out what you don’t!

22=-8+(6-1)d a 1 =-8a 6 =22 Arithmetic: Determine the explicit expression given the 1 st term and another term

Now Simplify, and solve for d 22=-8+(6- 1)d 22=-8+5d +8 30=5d 5 6=d Plug in what you know to write the explicit expression a n =-8+(n-1)6 Remember- This was given to begin with! Arithmetic: Determine the explicit expression given the 1 st term and another term

a n =-8+(n-1)6 Now we just need to simplify! a n =-8+6n-6 a n =6n-14

Arithmetic: Determine which term a value is given the explicit expression for the sequence What do we need to know?What do we already know? The n th term is 65 The explicit expression is a n =3n+2 What is n? What member of the sequence is 65? Plug in what you know to figure out what you don’t!

65=3n+2 63=3n =n 65 is the 21 st term in the sequence Arithmetic: Determine which term a value is given the explicit expression for the sequence

Geometric: Determine the explicit expression given the common difference and a term What do we need to know?What do we already know? The 4 th term is The common ration is +6. What is the first term? Plug in what you know to figure out what you don’t!

Geometric: Determine the explicit expression given the common difference and a term a 4 =1296 r=6 a n =a 1  r n =a 1  6 4-1

Geometric: Determine the explicit expression given the common difference and a term Now Simplify, and solve for a =a 1  =a 1  6 3 6=a =a 1  216 Remember- This was given to begin with! a n =6  6 n-1 Plug in what you know to write the explicit expression We just found this!

Geometric: Determine the explicit expression given the 1 st term and another term What do we need to know?What do we already know? The 1st term is 2. The 5 th term is What is the common ratio? Plug in what you know to figure out what you don’t!

Geometric: Determine the explicit expression given the 1 st term and another term a 1 =2a 5 =1250 a n =a 1  r n =2  r 5-1

Geometric: Determine the explicit expression given the 1 st term and another term 1250=2  r 5-1 Now Simplify, and solve for r 1250=2  r 4 625=r , 10, 50, 250, 1250 or 2, -10, 50, -250, 1250 a n =a 1  r n-1 a n =2  5 n-1 or a n =2  (-5) n-1