Answer :
Let's solve each part of the problem step by step:
1. Finding [tex]\( a_{15} \)[/tex] in the sequence defined by [tex]\( a_n = 3n + 1 \)[/tex]:
The formula for the sequence is [tex]\( a_n = 3n + 1 \)[/tex]. To find [tex]\( a_{15} \)[/tex], substitute [tex]\( n = 15 \)[/tex]:
[tex]\[
a_{15} = 3 \times 15 + 1 = 45 + 1 = 46
\][/tex]
So, [tex]\( a_{15} = 46 \)[/tex].
2. Finding [tex]\( a_4 \)[/tex] in the sequence defined by [tex]\( a_1 = 2; a_n = 4a_{n-1} - 3 \)[/tex]:
This is a recursive sequence. We are given [tex]\( a_1 = 2 \)[/tex] and the formula [tex]\( a_n = 4a_{n-1} - 3 \)[/tex]. Let's calculate the terms up to [tex]\( a_4 \)[/tex]:
- [tex]\( a_2 = 4 \times a_1 - 3 = 4 \times 2 - 3 = 8 - 3 = 5 \)[/tex]
- [tex]\( a_3 = 4 \times a_2 - 3 = 4 \times 5 - 3 = 20 - 3 = 17 \)[/tex]
- [tex]\( a_4 = 4 \times a_3 - 3 = 4 \times 17 - 3 = 68 - 3 = 65 \)[/tex]
Therefore, [tex]\( a_4 = 65 \)[/tex].
3. Finding [tex]\( a_8 \)[/tex] for the sequence [tex]\( 1, 4, 9, 16, \ldots \)[/tex]:
This sequence represents the squares of natural numbers: [tex]\( 1^2, 2^2, 3^2, \ldots \)[/tex]. To find [tex]\( a_8 \)[/tex], calculate [tex]\( 8^2 \)[/tex]:
[tex]\[
a_8 = 8^2 = 64
\][/tex]
So, [tex]\( a_8 = 64 \)[/tex].
4. Finding [tex]\( a_{12} \)[/tex] for the Fibonacci sequence [tex]\( 1, 1, 2, 3, 5, 8, \ldots \)[/tex]:
The Fibonacci sequence is defined as:
[tex]\[
a_1 = 1, \quad a_2 = 1, \quad a_n = a_{n-1} + a_{n-2}
\][/tex]
Calculate up to [tex]\( a_{12} \)[/tex]:
- [tex]\( a_3 = 1 + 1 = 2 \)[/tex]
- [tex]\( a_4 = 1 + 2 = 3 \)[/tex]
- [tex]\( a_5 = 2 + 3 = 5 \)[/tex]
- [tex]\( a_6 = 3 + 5 = 8 \)[/tex]
- [tex]\( a_7 = 5 + 8 = 13 \)[/tex]
- [tex]\( a_8 = 8 + 13 = 21 \)[/tex]
- [tex]\( a_9 = 13 + 21 = 34 \)[/tex]
- [tex]\( a_{10} = 21 + 34 = 55 \)[/tex]
- [tex]\( a_{11} = 34 + 55 = 89 \)[/tex]
- [tex]\( a_{12} = 55 + 89 = 144 \)[/tex]
Thus, [tex]\( a_{12} = 144 \)[/tex].
With these calculations, the descriptions match the values as follows:
- [tex]\( a_{15} \)[/tex] in the sequence defined by [tex]\( a_n = 3n + 1 \)[/tex] corresponds to 46.
- [tex]\( a_4 \)[/tex] for the sequence defined by [tex]\( a_1 = 2; a_n = 4a_{n-1} - 3 \)[/tex] corresponds to 65.
- [tex]\( a_8 \)[/tex] for the sequence [tex]\( 1,4,9,16,\ldots \)[/tex] corresponds to 64.
- [tex]\( a_{12} \)[/tex] for the Fibonacci sequence [tex]\( 1,1,2,3,5,8,\ldots \)[/tex] corresponds to 144.
1. Finding [tex]\( a_{15} \)[/tex] in the sequence defined by [tex]\( a_n = 3n + 1 \)[/tex]:
The formula for the sequence is [tex]\( a_n = 3n + 1 \)[/tex]. To find [tex]\( a_{15} \)[/tex], substitute [tex]\( n = 15 \)[/tex]:
[tex]\[
a_{15} = 3 \times 15 + 1 = 45 + 1 = 46
\][/tex]
So, [tex]\( a_{15} = 46 \)[/tex].
2. Finding [tex]\( a_4 \)[/tex] in the sequence defined by [tex]\( a_1 = 2; a_n = 4a_{n-1} - 3 \)[/tex]:
This is a recursive sequence. We are given [tex]\( a_1 = 2 \)[/tex] and the formula [tex]\( a_n = 4a_{n-1} - 3 \)[/tex]. Let's calculate the terms up to [tex]\( a_4 \)[/tex]:
- [tex]\( a_2 = 4 \times a_1 - 3 = 4 \times 2 - 3 = 8 - 3 = 5 \)[/tex]
- [tex]\( a_3 = 4 \times a_2 - 3 = 4 \times 5 - 3 = 20 - 3 = 17 \)[/tex]
- [tex]\( a_4 = 4 \times a_3 - 3 = 4 \times 17 - 3 = 68 - 3 = 65 \)[/tex]
Therefore, [tex]\( a_4 = 65 \)[/tex].
3. Finding [tex]\( a_8 \)[/tex] for the sequence [tex]\( 1, 4, 9, 16, \ldots \)[/tex]:
This sequence represents the squares of natural numbers: [tex]\( 1^2, 2^2, 3^2, \ldots \)[/tex]. To find [tex]\( a_8 \)[/tex], calculate [tex]\( 8^2 \)[/tex]:
[tex]\[
a_8 = 8^2 = 64
\][/tex]
So, [tex]\( a_8 = 64 \)[/tex].
4. Finding [tex]\( a_{12} \)[/tex] for the Fibonacci sequence [tex]\( 1, 1, 2, 3, 5, 8, \ldots \)[/tex]:
The Fibonacci sequence is defined as:
[tex]\[
a_1 = 1, \quad a_2 = 1, \quad a_n = a_{n-1} + a_{n-2}
\][/tex]
Calculate up to [tex]\( a_{12} \)[/tex]:
- [tex]\( a_3 = 1 + 1 = 2 \)[/tex]
- [tex]\( a_4 = 1 + 2 = 3 \)[/tex]
- [tex]\( a_5 = 2 + 3 = 5 \)[/tex]
- [tex]\( a_6 = 3 + 5 = 8 \)[/tex]
- [tex]\( a_7 = 5 + 8 = 13 \)[/tex]
- [tex]\( a_8 = 8 + 13 = 21 \)[/tex]
- [tex]\( a_9 = 13 + 21 = 34 \)[/tex]
- [tex]\( a_{10} = 21 + 34 = 55 \)[/tex]
- [tex]\( a_{11} = 34 + 55 = 89 \)[/tex]
- [tex]\( a_{12} = 55 + 89 = 144 \)[/tex]
Thus, [tex]\( a_{12} = 144 \)[/tex].
With these calculations, the descriptions match the values as follows:
- [tex]\( a_{15} \)[/tex] in the sequence defined by [tex]\( a_n = 3n + 1 \)[/tex] corresponds to 46.
- [tex]\( a_4 \)[/tex] for the sequence defined by [tex]\( a_1 = 2; a_n = 4a_{n-1} - 3 \)[/tex] corresponds to 65.
- [tex]\( a_8 \)[/tex] for the sequence [tex]\( 1,4,9,16,\ldots \)[/tex] corresponds to 64.
- [tex]\( a_{12} \)[/tex] for the Fibonacci sequence [tex]\( 1,1,2,3,5,8,\ldots \)[/tex] corresponds to 144.