Explanation

Given x + 1/x = 5

To find x² + 1/x²:

(x + 1/x)² = x² + 2(x)(1/x) + 1/x²

= x² + 2 + 1/x²

x² + 1/x² = (x + 1/x)² - 2

= 5² - 2

= 25 - 2

= 23

Explanation

Given:

A1 = 28 (first term)

d = -4 (common difference)

n = 7 (number of terms)

To find the nth term (an), use the formula:

an = a + (n-1)d

an = 28 + (7-1)(-4)

= 28 + 6(-4)

= 28 - 24

= 4

Explanation

 1 + 2 + 3 + ... + n = 15  

This is an arithmetic series. Use the formula:

n(n + 1)/2 = 15  

→ Multiply both sides by 2:  

n(n + 1) = 30  

Try n = 5:  

5(6) = 30 

So, n = 5

Now calculate:  

1³ + 2³ + 3³ + ... + n³  

Use formula:  

[n(n + 1)/2]² = [5×6/2]² = (15)² = 225

Explanation

x³ + 1/x³ = (x + 1/x)³ − 3(x + 1/x)  

= 3³ − 3×3

= 27 − 9

= 18

Explanation

 To solve the differential equation dy/dx = cosx, we integrate both sides with respect to x.

∫dy = ∫cosx dx

y = sinx + C

where C is the constant of integration.

Explanation

i^25 = i^(24+1) = i^1 = i (since i^4 = 1)

(1/i)^32 = (i^-1)^32 = i^(-32) = i^0 = 1 (since i^4 = 1)

So, (i^25 - (1/i)^32)^2

= (i - 1)^2

= i^2 - 2i + 1

= -1 - 2i + 1

= -2i

Explanation

An equation of the form ax² + bx + c = 0 is called a quadratic equation.

Here, it has no x-term, but still fits that form.

Explanation

 We are given:

To find $q$, divide both sides by 0.87:

Explanation

x + 1/x = 15

(x + 1/x)² = 15²

x² + 2(x)(1/x) + 1/x² = 225

x² + 2 + 1/x² = 225

x² + 1/x² = 225 - 2

x² + 1/x² = 223

Explanation

A surd is an irrational root.

√16 = 4 (rational)

√25 = 5 (rational)

√16 + √25 = 4 + 5 = 9 (rational)

√21 is irrational, so it's a surd.