Explanation

Aₙ = a + n × d, where d = common difference  

Here, A₁ = a + d, A₂ = a + 2d, ..., b = a + (n+1)d

So,  

Aₙ = a + n × [(b − a)/(n + 1)]

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

Let original number = 100

Increase 30% ⇒ 130

Decrease 10% ⇒ 130 - 13 = 117

Net change = 117 - 100 = 17% increase

Explanation

tan A + tan B = √2 + 1

Explanation

x³ + 1/x³ = (x + 1/x)³ − 3(x + 1/x)  

= 3³ − 3×3

= 27 − 9

= 18

Explanation

We know:

81 = 3⁴  

So,  

log₃ 81 = log₃ (3⁴) = 4

Explanation

 The continuity equation in fluid dynamics states that the product of cross-sectional area and velocity remains constant for incompressible fluids.

So, in deeper (wider) parts of a river or pipe, the velocity of water is lower — hence the phrase:

"Deep water runs still."

Explanation

 Leading zeros (before the first non-zero digit) are not significant.

In 0.000801, the digits 8, 0, and 1 are significant.

So, it has 3 significant figures

Explanation

In Flutter, the Scaffold() widget is typically used within the build() method of a StatefulWidget or StatelessWidget.

More specifically, it serves as the root widget for most screens in a Flutter application, providing the basic layout structure for Material Design. 

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.