Determining $a_7$ with Given Conditions

  • MHB
  • Thread starter anemone
  • Start date
  • Tags
    Conditions
In summary, $a_7$ is determined by using the given conditions and applying the formula for finding the general term of a sequence. The given conditions can vary, but typically include information about the first term and the common difference between terms in the sequence. Without these conditions, $a_7$ cannot be determined. It is possible for $a_7$ to be a negative number, depending on the given conditions and formula used. The value of $a_7$ is significant as it represents the 7th term in the sequence and can provide insight and aid in solving related problems.
  • #1
anemone
Gold Member
MHB
POTW Director
3,883
115
The positive integers $a_1,\,a_2,\,\cdots,a_7$ satisfy the conditions $a_6=144$, $a_{n+3}=a_{n+2}(a_{n+1}+a_n)$, where $n=1,\,2,\,3,\,4$.

Determine $a_7$.
 
Mathematics news on Phys.org
  • #2
anemone said:
The positive integers $a_1,\,a_2,\,\cdots,a_7$ satisfy the conditions $a_6=144$, $a_{n+3}=a_{n+2}(a_{n+1}+a_n)$, where $n=1,\,2,\,3,\,4$.

Determine $a_7$.

144 * 24 or 3456

as
$a_4 = a_3 ( a_2 + a_1) $

$a_5 = a_4 ( a_3 + a_2) $
= $a_3 ( a_2 + a_1) ( a_3 + a_2)$

$a_6 = a_5(a_4+a_3) $
= $a_3 ( a_2 + a_1) ( a_3 + a_2)( a_3 ( a_2 + a_1) + a_3))$
= $a_3^2 ( a_2 + a_1) ( a_3 + a_2)( a_3 ( a_2 + a_1+1)$

it is product of 5 numbers of which 2 are same and 2 differ by 1

we have 144 = 2^2 * 3 * 3 * 4

so $a_3= 2$, $a_2 = 1 $ and $a_1 = 2$
using relation above we can compute $a_4= 6$, $a_5 = 18 $ and $a_6 =144$ and $a_7$ = 144* 24
 
Last edited by a moderator:

FAQ: Determining $a_7$ with Given Conditions

1. How is $a_7$ determined?

$a_7$ is determined by using the given conditions and applying the formula for finding the general term of a sequence. This formula is typically given as $a_n = a_1 + (n-1)d$, where $a_n$ is the $n$th term of the sequence, $a_1$ is the first term, and $d$ is the common difference between terms. By plugging in the value of $n=7$ and the given conditions, $a_7$ can be calculated.

2. What are the given conditions for determining $a_7$?

The given conditions for determining $a_7$ can vary, but typically include information about the first term and the common difference between terms in the sequence. For example, the conditions may state that the first term is 3 and the common difference is 5. These conditions are essential for using the formula to find the value of $a_7$.

3. Can $a_7$ be determined if the conditions are not given?

No, $a_7$ cannot be determined without the given conditions. The conditions provide necessary information about the sequence in order to calculate the value of $a_7$. Without this information, the value of $a_7$ cannot be determined.

4. Is it possible for $a_7$ to be a negative number?

Yes, it is possible for $a_7$ to be a negative number. This depends on the given conditions and the formula used to calculate the value of $a_7$. For example, if the first term is a negative number and the common difference is also negative, then $a_7$ may be a negative number.

5. What is the significance of finding the value of $a_7$?

The value of $a_7$ is significant because it represents the 7th term in the sequence. This can provide insight into the pattern and behavior of the sequence, and can be used to make predictions about future terms in the sequence. Additionally, knowing the value of $a_7$ can help in solving other problems or equations related to the sequence.

Similar threads

Replies
1
Views
1K
Replies
1
Views
986
Replies
4
Views
2K
Replies
1
Views
961
Replies
3
Views
1K
Replies
2
Views
1K
Back
Top