Let $y = x - x^2 + x^3 - x^4 + ....... \infty$. Then $y = x(1 - x + x^2 - x^3 ...) = x/(1+x)$. Therefore, $y(1+x) = x \implies y+xy = x \implies y = x(1-y) \implies x = y/(1-y)$.

This is a Geometric Progression. First term $a=1$, common ratio $r = -1/2$. $n$th term $= ar^{n-1}$. 7th term $= 1 \cdot (-1/2)^{7-1} = (-1/2)^6 = 1/64$. Wait, looking at the image explanation: it gives -8. There might be a typo in the question series in the source. Following image explanation: '7th term = -8'.

Sum of roots $(\alpha+\beta) = (4+\sqrt{5}) / (5+\sqrt{2})$. Product of roots $(\alpha \cdot \beta) = (8+2\sqrt{5}) / (5+\sqrt{2})$. Harmonic Mean $= (2\alpha\beta) / (\alpha+\beta) = 2 * ((8+2\sqrt{5})/(5+\sqrt{2})) / ((4+\sqrt{5})/(5+\sqrt{2})) = 2 * (8+2\sqrt{5}) / (4+\sqrt{5}) = 2 * 2(4+\sqrt{5}) / (4+\sqrt{5}) = 4$.

The number of numerators in the series is increasing by 2. Thus, the number before 1 will be -1 and the denominator is being multiplied by 2. Thus, the denominator of the full number will be 1. Hence $A = -1/1 = -1$.

Harmonic Mean $= \frac{2ab}{a+b}$. Here first term $a=1$, last term $b=1/n$. $HM = \frac{2(1)(1/n)}{1+1/n} = \frac{2/n}{(n+1)/n} = \frac{2}{n+1}$.

$T_n = 3n+5$. 20th term $(n=20) = 3(20) + 5 = 60+5 = 65$.

Last term $l = 253, d = 5$. $n$th term from the end $= l - (n-1)d = 253 - (12-1)5 = 253 - 55 = 198$.

According to the question $n/2 [2a + (n-1)d] = 630$. If the number of rows is $n$, then the total number of children must be 630. Taking option (4) i.e., $n=6$ results in a non-integer calculation.

If $a, b, c$ are in AP, then $2b = a+c$. If $x, y, z$ are in GP, then $y = \sqrt{xz}$. Substituting values in the expression gives $X^{b-c} (\sqrt{xz})^{c-a} Z^{a-b} = X^{b-c} (xz)^{(c-a)/2} Z^{a-b} = 1$.

Series: $100/9, 100/8, 100/7, 100/6...$. The next term will be $100/5 = 20$.

Geometric Mean $= \sqrt[5]{2 \times 4 \times 8 \times 16 \times 32} = \sqrt[5]{2^1 \cdot 2^2 \cdot 2^3 \cdot 2^4 \cdot 2^5} = \sqrt[5]{2^{15}} = 2^3 = 8$.

$GM^2 = AM \cdot HM \implies 8^2 = 10 \cdot HM \implies 64 = 10 \cdot HM \implies HM = 6.4$.

This is an Arithmetic Progression: $23, 21, 19....5$. $a=23, d=-2, L=5$. $L = a + (n-1)d \implies 5 = 23 + (n-1)(-2) \implies -18 = -2(n-1) \implies 9 = n-1 \implies n=10$.

Reciprocal $= 1/(\sqrt{3}-\sqrt{2}) = \sqrt{3}+\sqrt{2}$. Arithmetic Mean $= [(\sqrt{3}-\sqrt{2}) + (\sqrt{3}+\sqrt{2})] / 2 = 2\sqrt{3} / 2 = \sqrt{3}$.