Sequences & Series

Edexcel IAL WMA12/01/P2/Oct 2022/Q3 (Sequences & Series, Periodic, Sigma Notation)

A sequence a1, a2, a3, … is defined by​​ 

an =cos2⁡nπ3 

Find the exact values of​​ 

• a1​​ 

• a2​​ 

• a3​​ 

(3)​​ 

• Hence find the exact value of

∑n=150n+cos2⁡nπ3

You must make your method clear.​​ 

(4)

SOLUTION​​ 

a-​​ 

(This is a simple question, you have to substitute the values of n in the given expression and get the value of the respectuve term. But here most of the students do the same mistake as they usually do in the trignometric identities​​ chapter. They get confused with​​ cos2⁡θ,​​ cos⁡θ2, and​​ cos⁡θ2. ​​​​ So remember,​​ cos⁡θ=cos⁡θ2≠cos⁡θ2).​​ 

i- For​​ n=1,

a1=cos⁡π32=122=14

ii- For​​ n=2,

a2=cos⁡2π32=-122 =14

iii- For​​ n=3

a3=cos⁡π2=-12=1 

b- On expanding the summation of the expression, the series would look like the following.

∑n=150n+cos2⁡nπ3=1+cos2⁡1×π3+2+cos2⁡2×π3+3+cos2⁡3×π3+4+cos2⁡4×π3+5+cos2⁡5×π3+6+cos2⁡6×π3

∑n=150n+cos2⁡nπ3=1+14+2+14+3+1+4+14+5+14+6+1

By this, it can be seen that​​ ∑n=150cos2⁡nπ3​​ is an alternating series or a repeating series. It doesn't fit into standard arithmetic or geometric progression series. Whereas,​​ ∑n=150n​​ is the arithmetic series with a common difference of 1. So, will split the expression into two separate series and apply their respective methods to sum them. And, at last, we are going to sum both the series as given in the question.​​ 

∑n=150n+cos2⁡nπ3=∑n=150n+ ∑n=150cos2⁡nπ3

∑n=150n=1+2+3+…+50

Its th arithmetic series with a common difference 1 and first term (a) as 1. So using the formula of sum of arithmetic series.​​ 

Using the second formula.​​ 

S50=502 1+50=25 x 51

S50=1275

∑n=150cos2⁡nπ3=14+14+1+14+14+1…….

The pattern repeats after every three terms: two​​ 14​​ followed by one 1. To find the sum, we can first determine how many complete cycles of the pattern there are in 50 terms and then calculate the sum accordingly.

Number of complete cycles in 50 terms =​​ 503= 16.666…=16 cycles

This complete cycle makes upto 48 terms of the series, and the remaining two terms of the series are​​ 0.25 &​​ 0.25.​​ 

Each cycle has a sum of​​ 0.25+0.25+1=1.5.

Now, we can calculate the sum of the first 16 complete cycles (that makes upto 48 terms of series) terms and the remaining two terms the sum as follow

Sum=Number of complete cycles×Sum of one cycle+49th term+50th term=(16×1.5)+14+14

=16×1.5+14+14=24+0.5=492

∑n=150n+cos2⁡nπ3=1275+492=25992

So, the sum of 50 terms of the given series is​​ 492