Question

complete the series by replacing 1, 10, ___ 52 ,85 ,126​

Answer 1

Answer:

1, 10, 27, 52 ,85, 126...

Explanation:

As, a(n) = (4n - 3)n or decagonal numbers series.

Answer 2

Answer:

Step-by-step explanation:

Concept:

Decagonal number:

The idea of triangle and square numbers is extended to the decagon by a figurate number called a decagonal number (a ten-sided polygon). However, the patterns used to create decagonal numbers are not rotationally symmetrical like those used to create triangle and square numbers. The ith decagon in the pattern has sides consisting of I dots spaced one unit apart from one another, and the nth decagonal number specifically counts the number of dots in the pattern of n nested decagons that all share a common corner. The formula below yields the n-th decagonal number.

$D n = 4n^2 - 3n.$

The initial group of decagonal numbers is:

0, 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451, 540, 637, 742, 855, 976, 1105, 1242, 1387, 1540, 1701, 1870, 2047, 2232, 2425, 2626, 2835, 3052, 3277, 3510, 3751, 4000, 4257, 4522, 4795, 5076, 5365, 5662, 5967, 6280, 6601, 6930, 7267, 7612, 7965, 8326 (sequence A001107 in the OEIS)

The nth decagonal number can also be determined by multiplying the nth pronic number by three times the square of n, or, to use algebraic notation, as

$Dn=n^ 2+3(n^ 2-n)$

Given:

1, 10, ___ 52 ,85 ,126​.

Find:

complete the series by replacing 1, 10, ___ 52 ,85 ,126​.

Solution:

given that complete the series by replacing 1, 10, ___ 52 ,85 ,126​.

The formula for the nth decagonal number is

$Dn=4n^2-3n$

Substitute $n=1,2,3,4,5,6.$

$D1=4(1)^2-3(1)=1$

$D2=4(2)^2-3(2)=10$

$D3=4(3)^2-3(3)=27$

$D4=4(4)^2-3(4)=52$

$D5=4(5)^2-3(5)=85$

$D6=4(6)^2-3(6)=126$

Hence the series 1,10,27,52,85,126.

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