sequences topic

Question

a company sales is described as Sn=2S_n-1 +S_n-2 +1 units , S_n represents the n^th sale. if the company experienced a jump in a sale by 70 units between 4th and 5th sale .how many will it sell in the 6th sale if s₃ = 20units

 

(a)120

(b)180

(c)198

(d)288

Answer 1

Since they're giving you the difference between the 4th and 5th sales, let's see if we can use the formula to find an expression for what we would expect that difference to be.

 

Based on the formula, S₅ = 2S₄ + S₃ + 1

 

That means that S₅ - S₄ = 2S₄ + S₃ + 1 - S₄ = S₄ + S₃ + 1

 

Since they tell us that S₅ - S₄ is 70, that means we can say that S₄ + S₃ + 1 = 70

 

But we also know that S₃ = 20, so we can plug that in and get S₄ + 20 + 1 = 70.  Or S₄ + 21 = 70.  We can solve for S₄:

 

S₄ + 21 = 70

S₄ = 70 -21 = 49.

 

And since was a jump of 70 from S₄ to S₅, That means that S₅ is 49 + 70 = 119.

 

Now we have everything we need to figure out S₆:

 

S₆ = 2S₅ +S₄+ 1 = 2 * 119 + 70 + 1 = 309