# Tutor profile: Harshdeep K.

## Questions

### Subject: Pre-Calculus

Does the infinite geometric series converge or diverge? $$1+\frac{5}{2}+\frac{25}{4}+\frac{125}{8}+ ...$$ If the series converges, find its sum.

To determine if the Geometric Progression series converges or diverges, find the common ratio r. Find the common ratio r. Take the ratio of the first two terms. $$r=\frac{\frac{5}{2}}{1}$$ Since the series is geometric, the ratio between any two consecutive terms is $$\frac{5}{2}$$ . Determine if the series converges or diverges. Take the absolute value of r. $$|r|=\frac{5}{2}\geq1$$ Since |r|≥1, the series diverges. Its sum does not exist

### Subject: Basic Math

The length and breadth of a rectangular courtyard is $$75\ m$$ and $$32\ m$$. Find the cost of leveling it at the rate of $$$3$$ per $$m^2$$. Also, find the distance covered by a boy to take 4 rounds of the courtyard.

Length of the courtyard = $$75\ m$$ Breadth of the courtyard = $$32\ m $$ Perimeter of the courtyard = $$2 \times (75 + 32) m$$ = $$2 \times 107\ m$$ = $$214\ m$$ Distance covered by the boy in taking 4 rounds = 4 × perimeter of courtyard = $$4 \times 214$$ = $$856\ m$$ We know that area of the courtyard = length × breadth = $$75 × 32\ m^2$$ = $$2400\ m^2$$ For $$1\ m^2$$, the cost of levelling = $$$3$$ For $$2400\ m^2$$, the cost of levelling = $$$3 × 2400$$ = $$$7200$$

### Subject: Calculus

If f(x) and g(x) are differentiable functions such that $$f '(x) = 7x^2\ and \ g '(x) = 5 x ^3$$ then the limit $$\lim_{x\to2}\frac{(f(x) + g(x)) - (f(2) + g(2)) }{ (x - 2) }$$ is equal to

Solution: $$\lim_{x\to1}\frac{(f(x) + g(x)) - (f(2) + g(2)) }{ (x - 2) }$$ The given limit is the definition of the derivative of $$f(x) + g(x)\ at\ x = 2$$. The derivative of the sum is equal to the sum of the derivatives. Hence the given limit is equal to$$ f '(2) + g '(2) = (7*(2)^2) + (5*(2)^3)=28+40=68$$

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