Question

2 4 4 If A = 2 un 5 4 then find A-1 2 un 5 3 f. 11​

Answer 1

Explanation:

Suppose we have the sum of a constant times k. What does this give us? For example,

X

4

k=1

3k = (3 × 1) + (3 × 2) + (3 × 3) + (3 × 4)

= 3 × (1 + 2 + 3 + 4)

= 3 × 10

= 30 .

But we can see from this calculation that the result also equals

3 × (1 + 2 + 3 + 4) = 3X

4

k=1

k ,

so that

X

4

k=1

3k = 3X

4

k=1

k .

In general, we can say that

Xn

k=1

ck = (c × 1) + (c × 2) + . . . + (c × n)

= c × (1 + ... + n)

= c

Xn

k=1

k .

Suppose we have the sum of k plus a constant. What does this give us? For example,

X

4

k=1

(k + 2) = (1 + 2) + (2 + 2) + (3 + 2) + (4 + 2)

= (1 + 2 + 3 + 4) + (4 × 2)

= 10 + 8

= 18 .

But we can see from this calculation that the result also equals

(4 × 2) + (1 + 2 + 3 + 4) = (4 × 2) +X

4

k=1

k ,

so that

X

4

k=1

(k + 2) = (4 × 2) +X

4

k=1

k .

In general, we can say that

Xn

k=1

(k + c) = (1 + c) + (2 + c) + . . . + (n + c)

= (c + c + . . . + c)

| {z }

n times

+ (1 + 2 + . . . + n)

= nc +

Xn

k=1

k .

www.mathcentre.ac.uk 7