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# Using recursive formulas of geometric sequences

Sal finds the 4th term in the sequence whose recursive formula is a(1)=-⅛, a(i)=2a(i-1).

## Want to join the conversation?

• I wonder what the logic is at ; how did Sal generalize into a(i)=(-1/8)(2^(i-1))?
• Another way to think of it is that every time you need a new term, you multiply by 2. If you have an original number of 3, your term numbers ` i ` would look like this top row. The row below would be your values:
` 1 2 3 4 5 6 `
` 3 6 12 24 48 96 `
You can write a quick, general formula from this for all geometric sequences:
`first value` x `multiplier` raised to `number of the term, minus one `
So it is `a(i) = a(1) ∙ (2) ^ (i - 1)`
If you don't adjust the exponent by one, you will find terms that are in the wrong location.

This is sometimes called the explicit formula, because you can generate any term if you know the first value and multiplier (common ratio).
In this simplified case I showed above, a(1) is 3
If we want to find the 4th term, here is how we calculate it:
a(4) = 3 ∙ (2) ^ (4 - 1)
or a(4) = 3 ∙ (2) ^ 3
or `a(4) = 3 ∙ (2)³ `
That means the 4th term is 3 ∙ 8 or 24
Bingo!

So for Sal's example, the terms are messier and we start out knowing only the first value and the multiplier, and the important information that it follows the rules for a geometric sequence.
` 1 2 3 4 5 6 `
` - 1/8 ? ? ? ? ? `
Each term is 2 times the previous. So I can reuse most of my equation from my simple example: `a(i) = a(1) ∙ (2) ^ (i - 1)`
For the fourth term
a(4) = - ⅛ ∙ (2) ^ (4 - 1)
a(4) = - ⅛ ∙ (2) ^ (4 - 1)
`a(4) = - ⅛∙ (2)³ `
which is a(4) = - ⅛∙ 8
That ends up with `-1 ` for the 4th term. Hope that helps
• Why is [Sal] Using "i" for the "current value" instead of using "n"?
• The most common variables used for indices are i, j, k, m, n. It does not always have to be n. In this case it is i. You will see other examples here on Khan that use j, k and m as well.
• He is putting the power inside the parentheses again, is he doing it right?
• Since 2 is positive, it does not matter if it is inside or outside
It would be better if he stayed consistent, but they are the same
the main reason you put it on the outside is if the number inside is negative which does make a difference
(-2)^2 is not the same as -2^2
Hope this helps
• Shouldn't you put the exponent always outside of the parenthesis when solving?
• In the case above, yes. But more generically, it depends on what you're multiplying. Note that
``(-4)^2 = -4 * -4 => 16``
is not equal to
``(-4^2) = -1 * (4*4) => -16``
• Is there 1 specific formula you use in geometric sequences?
• I know I'm a little late, but the formula is actually aₙ = a₁ * rⁿ ⁻ ¹.
(1 vote)
• So if a geometric series is the sum of the terms of the sequences, then Arithmetic series are the same. [correct me if am wrong]
• An arithmetic sequence is a sequence with the difference between two consecutive terms constant. The difference is called the common difference. Recall a difference is subtraction so (x2-x1).
Example: 1, 5, 9, 13, 17, ... where the common difference is 4

A geometric sequence is a sequence with the ratio between two consecutive terms constant. This ratio is called the common ratio. Recall a ratio is x2/x1.
Example: 1, 2, 4, 8, 16, 32, ...
Where the common ratio is 2.
• lol he said pause the video to find it out but that is why I am watching it XD I don't know how
• isnt there any faster way to do it instead of just solving each expression to find a(n)
• For a geometric sequence with recurrence of the form a(n)=ra(n-1) where r is constant, each term is r times the previous term.

This implies that to get from the first term to the nth term, we need to multiply by n-1 factors of r.

Therefore, for a geometric sequence, we can calculate a(n) explicitly by using a(n)=r^(n-1)*a(1).
(1 vote)
• After watching a lot of these videos, it seems like sequences are functions. Is this true?
• Yes this is because a sequence has 1st, 2nd, 3rd, etc. terms, so it will not repeat.