# How do you write the first six terms of the sequence a_n=n^2+3?

Dec 3, 2016

$4 , 7 , 12 , 19 , 28 , 39$

#### Explanation:

Method 1 - Direct substitution

${a}_{1} = {1}^{2} + 3 = 1 + 3 = 4$

${a}_{2} = {2}^{2} + 3 = 4 + 3 = 7$

${a}_{3} = {3}^{2} + 3 = 9 + 3 = 12$

${a}_{4} = {4}^{2} + 3 = 16 + 3 = 19$

${a}_{5} = {5}^{2} + 3 = 25 + 3 = 28$

${a}_{6} = {6}^{2} + 3 = 36 + 3 = 39$

$\textcolor{w h i t e}{}$
Method 2 - Differences

Since the formula is a quadratic one, its coefficients will be determined by the first $3$ terms.

Use direct substitution to write down the first $3$ terms:

$4 \textcolor{w h i t e}{00000} 7 \textcolor{w h i t e}{0000} 12$

Under the gaps between the terms write down the sequence of differences:

$4 \textcolor{w h i t e}{00000} 7 \textcolor{w h i t e}{0000} 12$
$\textcolor{w h i t e}{000} 3 \textcolor{w h i t e}{00000} 5$

Under the gap between the two terms, write the difference:

$4 \textcolor{w h i t e}{00000} 7 \textcolor{w h i t e}{0000} 12$
$\textcolor{w h i t e}{000} 3 \textcolor{w h i t e}{00000} 5$
$\textcolor{w h i t e}{000000} 2$

To this last line, add as many copies of the final difference as you would like extra terms of the original sequence:

$4 \textcolor{w h i t e}{00000} 7 \textcolor{w h i t e}{0000} 12$
$\textcolor{w h i t e}{000} 3 \textcolor{w h i t e}{00000} 5$
$\textcolor{w h i t e}{000000} 2 \textcolor{w h i t e}{00000} \textcolor{red}{2} \textcolor{w h i t e}{00000} \textcolor{red}{2} \textcolor{w h i t e}{00000} \textcolor{red}{2}$

Fill in extra terms on the line above by adding the differences:

$4 \textcolor{w h i t e}{00000} 7 \textcolor{w h i t e}{0000} 12$
$\textcolor{w h i t e}{000} 3 \textcolor{w h i t e}{00000} 5 \textcolor{w h i t e}{00000} \textcolor{red}{7} \textcolor{w h i t e}{00000} \textcolor{red}{9} \textcolor{w h i t e}{0000} \textcolor{red}{11}$
$\textcolor{w h i t e}{000000} 2 \textcolor{w h i t e}{00000} \textcolor{red}{2} \textcolor{w h i t e}{00000} \textcolor{red}{2} \textcolor{w h i t e}{00000} \textcolor{red}{2}$

Fill in extra terms on the line above by adding the differences:

$4 \textcolor{w h i t e}{00000} 7 \textcolor{w h i t e}{0000} 12 \textcolor{w h i t e}{0000} \textcolor{red}{19} \textcolor{w h i t e}{0000} \textcolor{red}{28} \textcolor{w h i t e}{0000} \textcolor{red}{39}$
$\textcolor{w h i t e}{000} 3 \textcolor{w h i t e}{00000} 5 \textcolor{w h i t e}{00000} \textcolor{red}{7} \textcolor{w h i t e}{00000} \textcolor{red}{9} \textcolor{w h i t e}{0000} \textcolor{red}{11}$
$\textcolor{w h i t e}{000000} 2 \textcolor{w h i t e}{00000} \textcolor{red}{2} \textcolor{w h i t e}{00000} \textcolor{red}{2} \textcolor{w h i t e}{00000} \textcolor{red}{2}$