Question

4,15, 40, 85, ?, 259 find missing term ​

Answer 1

Answer:

Formula:  $a(n) = n^3 + n^2 + n + 1.$

Step-by-step explanation:

$a(n) = n^3 + n^2 + n + 1.$

$a(1) = 1^3 + 1^2 + 1 + 1.\\a(1) = 1+1+1+1\\a(1) = 4$

$a(2) = 2^3 + 2^2 + 2 + 1.\\a(2) = 8+4+2+1\\a(2) = 15$

$a(3) = 3^3 + 3^2 + 3 + 1.\\a(3) = 27+9+3+1\\a(3) = 40$

$a(4) = 4^3 + 4^2 + 4 + 1.\\a(4) = 64+16+4+1\\a(4) = 85$

$a(5) = 5^3 + 5^2 + 5 + 1.\\a(4) = 125+25+5+1\\a(4) = 156$

$a(6) = 6^3 + 6^2 + 6 + 1.\\a(6) = 216+36+6+1\\a(6) =259$

Answer 2

Given series: $4,15, 40, 85, ?, 259$.

To find: the missing term.

Solution:

Understand that the given series follows the following pattern.

$\[a\left( n \right) = {n^3} + {n^2} + n + 1\]$

Observe the given series.

The first term is,

$\[\begin{array}{l} \Rightarrow a\left( 1 \right) = {1^3} + {1^2} + 1 + 1\\ \Rightarrow a\left( 1 \right) = 4\end{array}\]$

The second term is,

$\[\begin{array}{l} \Rightarrow a\left( 2 \right) = {2^3} + {2^2} + 2 + 1\\ \Rightarrow a\left( 2 \right) = 8 + 4 + 2 + 1\\ \Rightarrow a\left( 2 \right) = 15\end{array}\]$

The third term is,

$\[\begin{array}{l} \Rightarrow a\left( 3 \right) = {3^3} + {3^2} + 3 + 1\\ \Rightarrow a\left( 3 \right) = 27 + 9 + 3 + 1\\ \Rightarrow a\left( 3 \right) = 40\end{array}\]$

The fourth term is,

$\[\begin{array}{l} \Rightarrow a\left( 4 \right) = {4^3} + {4^2} + 4 + 1\\ \Rightarrow a\left( 4 \right) = 64 + 16 + 4 + 1\\ \Rightarrow a\left( 4 \right) = 85\end{array}\]$

Find the missing term.

$\[\begin{array}{l} \Rightarrow a\left( 5 \right) = {5^3} + {5^2} + 5 + 1\\ \Rightarrow a\left( 5 \right) = 125 + 25 + 5 + 1\\ \Rightarrow a\left( 5 \right) = 156\end{array}\]$

Hence, the missing term is 156.