The sum, $$\(S_{n}\)$$, of the first $$\(n\)$$ terms of an arithmetic progression is given by $$\[ S_{n}=n^{2}+4 n . \]$$ The $$\(k\)$$ th term in the progression is greater than 200. Find the smallest possible value of $$\(k\)$$. ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................ ............................................................................................................................................................
Exam No：9709_w20_qp_12 Year：2020 Question No：4
Answer:
$\begin{array}{l}S_{x} \text { and } S_{x+1} \\ a=5, d=2\end{array}$
$a+(n-1) d>200 \rightarrow 5+2(k-1)>200$
$(k=) 99$
Alternative method for question 4
$\begin{array}{l}\frac{n}{2}(2 a+(n-1) d) \equiv n^{2}+4 n \rightarrow\left(\frac{d}{2}=1, a-\frac{1}{2} d=4\right) \\ d=2, a=5 \\ a+(n-1) d>200 \rightarrow 5+2(k-1)>200\end{array}$
$(k=) 99$
Alternative method for question 4
$\begin{array}{l}\operatorname{sum}_{k}-\operatorname{sum}_{k-1} \rightarrow k^{2}+4 k-(k-1)^{2}-4(k-1) \\ 2 k+3>200 \text { or }=200\end{array}$
$(k=) 99$
$a+(n-1) d>200 \rightarrow 5+2(k-1)>200$
$(k=) 99$
Alternative method for question 4
$\begin{array}{l}\frac{n}{2}(2 a+(n-1) d) \equiv n^{2}+4 n \rightarrow\left(\frac{d}{2}=1, a-\frac{1}{2} d=4\right) \\ d=2, a=5 \\ a+(n-1) d>200 \rightarrow 5+2(k-1)>200\end{array}$
$(k=) 99$
Alternative method for question 4
$\begin{array}{l}\operatorname{sum}_{k}-\operatorname{sum}_{k-1} \rightarrow k^{2}+4 k-(k-1)^{2}-4(k-1) \\ 2 k+3>200 \text { or }=200\end{array}$
$(k=) 99$
Knowledge points:
1.6.3 use the formulae for the nth term and for the sum of the first n terms to solve problems involving arithmetic or geometric progressions (Including knowledge that numbers a,b,c are 'in arithmetic progression' if 2 b=a+c (or equivalent) and are 'in geometric progression' if (or equivalent) (Questions may involve more than one progression.)
Solution:
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