Arithmetic Sequences II: Series
IB Mathematics AA · SL & HL · Shadow Worksheet
Practice
Question 1
Find the sum of the following arithmetic series:
(a)\(3+7+11+\cdots\) to 20 terms
(b)\(50+46+42+\cdots+(-10)\)
(c)\(\displaystyle\sum_{r=1}^{15}(3r-2)\)
(a)
\(a=3,d=4,n=20\)
\(S_{20}= frac{20}{2}(6+76)\)
(b)
Find \(n\)
\(u_n=-10=50+(n-1)(-4)\Rightarrow n=16\)
Sum
\(S_{16}= frac{16}{2}(50+(-10))=8 imes40\)
(c)
\(a=1,d=3,n=15\)
\(S_{15}= frac{15}{2}(2+44)\)
Question 2
The sum of the first \(n\) terms of an arithmetic series is \(S_n = 3n^2 + 2n\).
(a)Find \(u_1\) and \(u_2\).
(b)Find the common difference.
(c)Find \(u_n\).
(a)
\(u_1=S_1,\;u_2=S_2-S_1\)
\(S_1=5,\;S_2=16\Rightarrow u_2=11\)
(c)
\(u_n=S_n-S_{n-1}\)
\((3n^2+2n)-(3(n-1)^2+2(n-1))=6n-1\)
Question 3
An arithmetic series has first term \(a\) and common difference \(d\). The 5th term is 17 and the sum of the first 10 terms is 155. Find \(a\) and \(d\).
Two equations
\(a+4d=17\quad(1)\); \( frac{10}{2}(2a+9d)=155\Rightarrow 2a+9d=31\quad(2)\)
Solve
\((2)-2 imes(1):\;d=-3\Rightarrow a=29\)
Question 4
Find the sum of all integers between 1 and 200 that are divisible by 7.
Sequence
\(7,14,21,\ldots,196\); \(n=28\)
Sum
\(S_{28}= frac{28}{2}(7+196)=14 imes203\)
Question 5
A theatre has 25 rows. The front row has 18 seats and each subsequent row has 2 more seats than the row in front. Find the total number of seats.
Arithmetic series
\(a=18,d=2,n=25\)
Sum
\(S_{25}= frac{25}{2}(36+48)= frac{25}{2} imes84\)
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