Observe the pattern and fill in the missing numbers.
10 × 12 = (11 – 1)(11 + 1) = $11^{2}$ – 1
11 × 13 = (12 – 1)(12 + 1) = $12^{2}$ – 1
12 × 14 = (13 – 1)(13 + 1) = $13^{2}$ – 1
13 × 15 = (– 1)( + 1) = – 1
16 × = (17 – 1)(17 + 1) = – 1

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Q. Identify the type of the following sequence:

\frac{1^{2}}{1} + \frac{5^{2}}{11} + \frac{9^{2}}{11^{2}} + \frac{13^{2}}{11^{3}} + . . . . . . .

Q. The sum of the series

1 + \frac{2}{11} + \frac{5}{11^{2}} + \frac{10}{11^{3}} + \frac{17}{11^{4}} + . .

Complete the table below:

$\begin{matrix} O u t p u t (U n i t s) & A v e r a g e C o s t (R s .) & M a r g i n a l C o s t (R s .) 1 & 12 & - 2 & 10 & - 3 & - & 10 4 & 10.5 & - 5 & 11 & - 6 & - & 17 \end{matrix}$

Q. Observe the following pattern

1^{2} = \frac{1}{6} [1 \times (1 + 1) \times (2 \times 1 + 1)] 1^{2} + 2^{2} = \frac{1}{6} [2 \times (2 + 1) \times (2 \times 2 + 1)] 1^{2} + 2^{2} + 3^{2} = \frac{1}{6} [3 \times (3 + 1) \times (2 \times 3 + 1)] 1^{2} + 2^{2} + 3^{2} + 4^{2} = \frac{1}{6} [4 \times (4 + 1) \times (2 \times 4 + 1)]

and find the values of each of the following:
(i) 1² + 2² + 3² + 4²+ ... + 10²
(ii) 5² + 6² + 7² + 8² + 9² + 10² + 11² + 12²

Q. Find the value of the given expression:

\frac{1}{4 - \sqrt{15}} - \frac{1}{\sqrt{15} - \sqrt{14}} + \frac{1}{\sqrt{14} - \sqrt{13}} - \frac{1}{\sqrt{13} - \sqrt{12}} + \frac{1}{\sqrt{12} - \sqrt{11}} - \frac{1}{\sqrt{11} - \sqrt{10}} + \frac{1}{\sqrt{10} - 3}

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Observe the pattern and fill in the missing numbers. 10 × 12 = (11 – 1)(11 + 1) = 112 – 1​ 11 × 13 = (12 – 1)(12 + 1)​ = 122 – 1 12 × 14 = (13 – 1)(13 + 1) = 132 – 1 13 × 15​ = (– 1)( + 1) = – 1​ 16 × = (17 – 1)(17 + 1) = – 1

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Observe the pattern and fill in the missing numbers.
10 × 12 = (11 – 1)(11 + 1) = $11^{2}$ – 1
11 × 13 = (12 – 1)(12 + 1) = $12^{2}$ – 1
12 × 14 = (13 – 1)(13 + 1) = $13^{2}$ – 1
13 × 15 = (– 1)( + 1) = – 1
16 × = (17 – 1)(17 + 1) = – 1