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The sum of the first n terms of a sequence is n^2 + 5n.  What is the 1000th term of the sequence?

 May 7, 2020
 #1
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The list of numbers is just even numbers starting at 6, so the 1000th term would be 2006.

 May 7, 2020
 #2
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The sequence is:
3, 6, 9, 12, 15, 18, 21 =n^2 + 5*n=7^2 + 35=84
Therefore the 1000th term =First term + 3*(1000 - 1)
                                                  =3 + (3*999)
                                                   =3 + 2,997
                                                   =3,000 - which is 1000th term.

 May 7, 2020
edited by Guest  May 7, 2020
 #3
avatar+932 
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The sum of the first term is just the first term. So if we plug in 1 to the equation, we'll find the first term which is (1^2) + 5 * 1 = 6. From this I can already see the difference in our solutions.

HELPMEEEEEEEEEEEEE  May 7, 2020
 #4
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But, doesn't he/she say "The sum of first n terms" =n^2 + 5n, which implies "several terms" summing up to n^2 + 5*n ?? Otherwise, he/she could have said " The sum of the FIRST term" =n^2 + 5n.

Guest May 7, 2020
 #5
avatar+24992 
+1

The sum of the first n terms of a sequence is \(n^2 + 5n\).  
What is the \(1000th\) term of the sequence?

 

Formula:  \(\mathbf{\boxed{s_n =\dfrac{(a_1+a_n)}{2}\cdot n }}\)

 

\(\begin{array}{|rcll|} \hline \mathbf{a_1} &=& \mathbf{s_1} \quad | \quad s_n =n^2 + 5n \\ &=& 1^2 + 5*1 \\ \mathbf{a_1} &=& \mathbf{6} \\ \hline \end{array} \)

 

\(\begin{array}{|rcll|} \hline s_n &=& \dfrac{(a_1+a_n)}{2}\cdot n \quad | \quad a_1=6,\ s_n = n^2 + 5n \\\\ n^2 + 5n &=& \dfrac{(6+a_n)}{2}\cdot n \\\\ 2n^2 + 10n &=& (6+a_n)n \quad | \quad :n \\ 2n + 10 &=& 6+a_n \\ 2n + 4 &=& a_n \\ \mathbf{a_n} &=& \mathbf{2n+4} \\ \hline \end{array}\)

 

\(\begin{array}{|rcll|} \hline \mathbf{a_n} &=& \mathbf{2n+4} \quad | \quad n=1000 \\ a_{1000} &=& 2*1000+4 \\ \mathbf{a_{1000}}&=& \mathbf{2004} \\ \hline \end{array}\)

 

laugh

 May 8, 2020

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