Sequences and Series Sum of 1s |
Sum of 1 n times
rth term: 1
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1 + 1 + 1 + 1 + … + 1 |
n |
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n |
Derivation of the Formula
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1 + 1 + 1 + 1 + … + 1 |
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n × 1 |
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n |
Proof by Induction
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n |
Check that the formula is correct when n = 1
LHS: |
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1 |
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RHS: |
n |
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1 |
Assume that the formula is true for n = k
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+ 1 |
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k |
+ 1 |
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(k + 1) |
If the formula is true for n = k then it is also true for n = (k + 1)
If the formula is true for n = k then it is true for n = (k + 1) and as it is true for n = 1, then by using mathematical induction the statement is true for all positive integers
Sum of the first n natural numbers
rth term: r
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1 + 2 + 3 + 4 + … + n |
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Derivation of the Formula
Let S |
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S |
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1 + 2 + 3 + … + (n-2) + (n-1) + n |
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S |
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n + (n-1) + (n-2) + … + 3 + 2 + 1 |
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2S |
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[1 + n] + [2 + (n-1)] + [3 + (n-2)] + … + [(n-2) + 3] + [(n-1) + 2] + [n + 1] |
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n × (n + 1) |
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S |
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Proof by Induction
The formula for the sum of the first n natural number
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Check that the formula is correct when n = 1
LHS: |
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1 |
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RHS: |
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1 |
Assume that the formula is true for n = k
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+ (k+1) |
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+ (k+1) |
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If the formula is true for n = k then it is also true for n = (k + 1)
If the formula is true for n = k then it is true for n = (k + 1) and as it is true for n = 1, then by using mathematical induction the statement is true for all positive integers
Sum of the first n square numbers
rth term: r2
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1 + 4 + 9 + 16 + … + n2 |
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Derivation of the Formula
Let S = |
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(23 - 13) + (33 - 23) + … + (n3 - (n-1)3) + ((n+1)3 - n3) |
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(n+1)3 - 1 |
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n3 + 3n2 + 3n |
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3S + |
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+ n |
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3S + |
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n3 + 3n2 + 3n |
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3S + |
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2n3 + 6n2 + 6n |
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6S + 3n2 + 5n |
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6S |
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2n3 + 6n2 + 6n - 3n2 - 5n |
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2n3 + 3n2 + n |
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n(n+1)(2n+1) |
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S |
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Proof by Induction
The formula for the sum of the first n squares number
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Check that the formula is correct when n = 1
LHS: |
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1 |
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RHS: |
n |
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1 |
Assume that the formula is true for n = k
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+ (k+1)2 |
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+ (k+1)2 |
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If the formula is true for n = k then it is also true for n = (k + 1)
If the formula is true for n = k then it is true for n = (k + 1) and as it is true for n = 1, then by using mathematical induction, the formula is true for all positive integers
Sum of the first n cubic numbers
rth term: r3
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1 + 8 + 27 + 64 + … + n3 |
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Derivation of the Formula
Let S = |
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= (24 - 14) + (34 - 24) + … + (n4 - (n-1)4) + ((n+1)4 - n4) |
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= (n+1)4 - 1 |
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= n4 + 4n3 + 6n2 + 4n |
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4S + |
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+ n |
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4S + n(n+1)(2n+1) + 2n(n+1) + n |
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4S + 2n3 + 5n2 + 4n |
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n4 + 4n3 + 6n2 + 4n |
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4S + 2n3 + 5n2 + 4n |
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4S |
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n4 + 2n3 + n2 |
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n2(n + 1)2 |
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S |
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Proof by Induction
The formula for the sum of the first n cubic number
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Check that the formula is correct when n = 1
LHS: |
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1 |
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RHS: |
n |
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1 |
Assume that the formula is true for n = k
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+ (k+1)2 |
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+ (k+1)3 |
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If the formula is true for n = k then it is also true for n = (k+1)
If the formula is true for n = k then it is true for n = (k + 1) and as it is true for n = 1, then by using mathematical induction, the formula is true for all positive integers
Sums of Polynomials
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Sum of the first n numbers |
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Sum of the first n square numbers |
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Sum of the first n cube numbers |
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