Advanced Starters

Progressing from Sequences

Maths is full of sequences, if it wasn’t there wouldn’t be any maths!

Formulae describe the sequences.

eg. the sequence 12,16,20,24, … starts with 12 and 4 is added to obtain the next term

Hence the sequence is given by the formula n^th term = 4n+8

Because the same number is added to obtain each successive term, the sequence forms an Arithmetic Progression.

For the example above 1^st term = 12 = 12

2^nd term = 16 = 12+(1x4)

3^rd term = 20 = 12+(2x4)

4^th term = 24 = 12+(3x4)

n^th term = 12+((n-1)x4)

Using the usual notation a = 1^st term

d = common difference

1A Write down a and d for each of these arithmetic progressions and use the formula

n^th term = a+(n-1)d to find the 47^th term of the progression.

1. 5,8,11,14,… 3. 7,14,21,28,… 5. –13,-8,-3,2,…

2. 6,17,28,39, … 4. 50,46,42,38,… 6. –4,-10,-16,-22,…

Again returning to the sequence/arithmetic progression 12,16,20,24, …

Sum of 1^st term = S₁ = 12 = a

Sum of 1^st 2 terms = S₂ = 12+16 = a+(a+d)

Sum of 1^st 3 terms = S₃ = 12+16+20 = a+(a+d)+(a+2d)

Sum of 1^st 4 terms = S₄ = 12+16+20+24 = a+(a+d)+(a+2d)+(a+3d)

Sum of 1^st 5 terms = S₅ = 12+16+20+24+28 = a+(a+d)+(a+2d)+(a+3d)+(a+4d)

1B Simplify each of the algebraic expressions above and show that

Sum of 1st n terms

Hint. Remember "triangular numbers". You probably thought you would never see them again!

1C Use the formula above to find:-

1. Sum of first 7 terms of the progression 5,7,9,11,…
2. Sum of first 8 terms of the progression 17,29,41,53,…
3. Sum of first 5 terms of the progression –9,-6,-3,0,…
4. Sum of first 10 terms of the progression 21,17,13,9,…
5. Sum of first 6 terms of the progression 7,4·2,1·4,-1·4,…

1D

1. How could you use the formula to find the sum of the 6^th to the 10^th terms inclusive of the sequence 8,15,22,29,36,… ?
2. Find a sequence such that the sum of the 4^th to the 8^th terms inclusive is 47.

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