Is this sequence 10, 15, 20, 25, 30 an arithmetic or geometric sequence? Then find the explicit formula and recursive formula for the given sequence.
Well, 10, 15, 20, 25, 30... I must say, this sequence seems quite predictable, like the lineup for a clown convention. It tickles my funny bone!
Now, getting back to your question, this sequence is an arithmetic sequence. Why, you ask? Well, because the difference between each term is a fixed amount of 5. It's like a clown car, the passengers are all having to squeeze closer together by 5 units!
The explicit formula for an arithmetic sequence is given by An = a + (n - 1)d, where 'a' is the first term and 'd' is the common difference. In this case, a is equal to 10, and the common difference 'd' is 5. So, the explicit formula for this sequence would be An = 10 + (n - 1)5. Now, that's some math that even a clown can juggle with!
As for the recursive formula... well, let's just say it's a bit trickier to explain. Imagine you have a clown chasing another clown, and each clown is 5 units apart. To catch up, the second clown needs to add 5 units to the previous term. So, the recursive formula for this sequence would be A1 = 10 and An = An-1 + 5. It's like a clown trying to catch up with the rest of the circus!
I hope that brings some laughter to your mathematical journey!
To determine whether the sequence 10, 15, 20, 25, 30 is arithmetic or geometric, we need to check if there is a common difference between consecutive terms.
Looking at the sequence, we can see that each term increases by 5. Therefore, this is an arithmetic sequence with a common difference of 5.
To find the explicit formula, we need to express each term in terms of its position in the sequence. Let's say the position of a term is denoted by "n". The first term (n = 1) is 10, and the common difference is 5. The explicit formula for an arithmetic sequence is:
term(n) = first term + (n - 1) * common difference
Plugging in the values, we have:
term(n) = 10 + (n - 1) * 5
For the recursive formula, we need to find a formula that depends on the previous term(s) in the sequence. In this case, since the previous term always adds 5 to the current term, we can express it as:
term(n) = term(n - 1) + 5
To summarize:
Explicit formula: term(n) = 10 + (n - 1) * 5
Recursive formula: term(n) = term(n - 1) + 5
To determine if a sequence is arithmetic or geometric, we need to look for patterns between consecutive terms.
In the given sequence 10, 15, 20, 25, 30, let's calculate the differences between consecutive terms:
15 - 10 = 5
20 - 15 = 5
25 - 20 = 5
30 - 25 = 5
Since the differences between consecutive terms (5) are constant, we can conclude that this sequence is an arithmetic sequence.
To find the explicit formula for an arithmetic sequence, we need the common difference (d) and the first term (a₁). In this case:
d = 5 (the constant difference between terms),
a₁ = 10 (the first term).
The explicit formula for an arithmetic sequence is given by:
aₙ = a₁ + (n - 1) * d
Substituting the given values into the explicit formula:
aₙ = 10 + (n - 1) * 5
aₙ = 10 + 5n - 5
aₙ = 5n + 5
So, the explicit formula for the given arithmetic sequence is aₙ = 5n + 5.
To find the recursive formula for an arithmetic sequence, we need the common difference (d) and the first term (a₁). In this case:
d = 5 (the constant difference between terms),
a₁ = 10 (the first term).
The recursive formula for an arithmetic sequence is given by:
aₙ = aₙ₋₁ + d
Substituting the given values into the recursive formula:
aₙ = aₙ₋₁ + 5
aₙ = aₙ₋₁ + 5
So, the recursive formula for the given arithmetic sequence is aₙ = aₙ₋₁ + 5.