(A) To climb an $n$-step staircase,the person can reach the $n$-th step either from the $(n-1)$-th step (by taking one step) or from the $(n-2)$-th st

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(A) To climb an $n$-step staircase,the person can reach the $n$-th step either from the $(n-1)$-th step (by taking one step) or from the $(n-2)$-th step (by taking two steps).Therefore,the recurrence relation is $C_n = C_{n-1} + C_{n-2}$ for $n \ge 2$.This is the definition of the Fibonacci sequence.Given the recurrence relation $C_n = C_{n-1} + C_{n-2}$,we can substitute $n = 20$ to get $C_{20} = C_{19} + C_{18}$.Thus,$C_{18} + C_{19} = C_{20}$.

Class 11 Mathematics Sequences and Series nth term of special series, Sum to n terms and Infinite number of terms

$A$ person wants to climb a $n$-step staircase using one step or two steps at a time. Let $C_n$ denote the number of ways of climbing the $n$-step staircase. Then $C_{18} + C_{19}$ equals

A
$C_{20}$
B
$C_{21}$
C
greater than $C_{21}$
D
less than $C_{20}$

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Sequences and Series

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nth term of special series, Sum to n terms and Infinite number of terms

If $[x]$ denotes the greatest integer less than or equal to $x$,then $\sum_{n=8}^{100} \left[ \frac{(-1)^{n} n}{2} \right]$ is equal to:

EasyJEE MAIN 2021

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$\sum\limits_{r = 1}^n {\sum\limits_{m = 1}^r {m} } = \dots$

Difficult

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The $n^{th}$ term of the following series $(1 \times 3) + (3 \times 5) + (5 \times 7) + (7 \times 9) + \dots$ will be

Easy

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Consider an infinite geometric series with first term $ a $ and common ratio $ r $. If the sum is $ 4 $ and the second term is $ \frac{3}{4} $,then find the values of $ a $ and $ r $.

DifficultKCET 2014

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The sum of the first $20$ terms of the series $1 + \frac{3}{2} + \frac{7}{4} + \frac{15}{8} + \frac{31}{16} + \dots$ is?

DifficultJEE MAIN 2018

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$A$ person wants to climb a $n$-step staircase using one step or two steps at a time. Let $C_n$ denote the number of ways of climbing the $n$-step staircase. Then $C_{18} + C_{19}$ equals

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If $[x]$ denotes the greatest integer less than or equal to $x$,then $\sum_{n=8}^{100} \left[ \frac{(-1)^{n} n}{2} \right]$ is equal to:

$\sum\limits_{r = 1}^n {\sum\limits_{m = 1}^r {m} } = \dots$

The $n^{th}$ term of the following series $(1 \times 3) + (3 \times 5) + (5 \times 7) + (7 \times 9) + \dots$ will be

Consider an infinite geometric series with first term $ a $ and common ratio $ r $. If the sum is $ 4 $ and the second term is $ \frac{3}{4} $,then find the values of $ a $ and $ r $.

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