The first term of the 11^{th} group in the fo | vedclass

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Question

(C) Observe the first term of each group:Group $1$: $1$Group $2$: $2$Group $3$: $5$Group $4$: $10$Group $5$: $17$Let $a_n$ be the first term of the $n

Vedclass · Accepted Answer

$101$

Vedclass · Answer

(C) Observe the first term of each group:Group $1$: $1$Group $2$: $2$Group $3$: $5$Group $4$: $10$Group $5$: $17$Let $a_n$ be the first term of the $n^{th}$ group.The sequence of first terms is $1, 2, 5, 10, 17, \dots$Calculate the differences between consecutive terms:$2 - 1 = 1$$5 - 2 = 3$$10 - 5 = 5$$17 - 10 = 7$The differences are $1, 3, 5, 7, \dots$, which is an arithmetic progression with common difference $2$.The $n^{th}$ term of this sequence is given by $a_n = a_1 + \sum_{k=1}^{n-1} (2k - 1) = 1 + (n-1)^2$.For the $11^{th}$ group, $n = 11$:$a_{11} = (11 - 1)^2 + 1 = 10^2 + 1 = 100 + 1 = 101$.

The first term of the $11^{th}$ group in the following sequence of groups $(1), (2, 3, 4), (5, 6, 7, 8, 9), \dots$ is:

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