More Complicated Sequence

The given sequence is an example of complex arithmetic sequence. A complex arithmetic sequence is an arithmetic sequence whose differences form another arithmetic sequence giving the second common difference. In the given sequence, we have to consider the components of this type of sequence. 1. First term of the given sequence 2. The common difference of the first two terms. 3. The number of terms needed. 4. The second common difference.

I have formulate this formula for this and it states like this:

${ a }_{ n }={ a }_{ 1 }+\left[ \frac { n-1 }{ 2 } \left( 2{ a }_{ x }+(n-2){ d }_{ 2 } \right) \right]$ where ${ a }_{ n }$ is the last term, ${ a }_{ 1 }$ is the first term of the given term, $n$ is the required number of terms , ${ a }_{ x }$ is the difference of the first two terms and ${ d }_{ 2 }$ is the second common difference.

Using the formula given, We all know the givens: ${ a }_{ 1 }$ =3 $n$ =10 ${ a }_{ x }$ = ${ a }_{ x }={ a }_{ 2 }-{ a }_{ 1 }=8-3=5$ ${ d }_{ 2 }=5$

Now the tenth term would be: ${ a }_{ n }={ a }_{ 1 }+\left[ \frac { n-1 }{ 2 } \left( 2{ a }_{ x }+(n-2){ d }_{ 2 } \right) \right] \\ { a }_{ n }=3+\left[ \frac { 10-1 }{ 2 } \left( 2(5)+(10-2)(5) \right) \right] \\ { a }_{ n }=3+\left[ \frac { 9 }{ 2 } (10+8(5)) \right] \\ \\ { a }_{ n }=3+\left[ \frac { 9 }{ 2 } \left( 10+40 \right) \right] \\ { a }_{ n }=3+\left[ \frac { 9 }{ 2 } \left( 50 \right) \right] \\ { a }_{ n }=3+\left[ 225 \right] \\ { a }_{ n }=228$

To check that the answer is correct: ${ a }_{ 1 }=3\\ { a }_{ 2 }=3+5=8\\ { a }_{ 3 }=8+5(2)=18\\ { a }_{ 4 }=18+5(3)=33\\ { a }_{ 5 }=33+5(4)=53\\ { a }_{ 6 }=53+5(5)=78\\ { a }_{ 7 }=78+5(6)=108\\ { a }_{ 8 }=108+5(7)=143\\ { a }_{ 9 }=143+5(8)=183\\ { a }_{ 10 }=183+5(9)=\boxed { 228 }$ .

$a_{n+1}=a_{n}+5 (n+1)$ and $a_{0}=3$ $a_{0+1}=a_{0}+5(0+1)=3+5=8$ and so on until $a_{8}=a_{7}+5 (8)=143+40=183$ $a_{9}=a_{8}+5 (9)=183+45=\boxed {228}$