Question

Find the positive integral value of n such that:

(1 × 2¹) + (2 × 2²) + (3 × 2³) + (n × 2ⁿ) = 2^(n + 10) + 2​

Answer 1

Answer:

$\huge \red{n = 513}$

Step-by-step explanation:

Let

$s = 1. {2}^{1} + 2 .{2}^{2} + 3. {2}^{2} + . \: . \: . \: . + n. {2}^{n} \\ \\ 2s = 1. {2}^{2} + 2. {2}^{3} + . \: . \: . \: . + n . {2}^{n + 1} \\ \\ subtract \: both \: equations \: \\ \\ 2s - s = - 1. {2}^{1} - \{2 {}^{2} + {2}^{3} + . \: . \: + {2}^{n} \} + n. {2}^{n + 1} \\ \\ s = - 2 + n. {2}^{n + 1} - \frac{2 {}^{2} ( {2}^{n - 1} - 1)}{2 - 1} \\ \\ s = - 2 + n. {2}^{n + 1} - {2}^{n + 1} + 2 {}^{2} \\ \\ \red{ \boxed{s = {2}^{n + 1} (n - 1) + 2}}$

Now given

$s = {2}^{n + 10} + 2 = {2}^{n + 1}(n - 1) + 2 \\ \\ \implies \: {2}^{n + 10} = {2}^{n + 1} (n - 1) \\ \\ \implies \: {2}^{n } . {2}^{10} = {2}^{n }.2 (n - 1) \\ \\ \implies \: {2}^{10} = 2(n - 1) \\ \\ \implies \: n - 1 = {2}^{9} = 512 \\ \\ \implies \: n = 512 + 1 = 513$

Answer 2

Question:-

Find the positive integral value of n such that

$\sf \large 1.2 {}^{1} + 2.2 {}^{2} + 3.2 {}^{3} + n.2 {}^{n} = 2 {}^{(n + 10)} + 2$

Given:-

$\sf \large 1.2 {}^{1} + 2.2 {}^{2} + 3.2 {}^{3} + n.2 {}^{n} = 2 {}^{(n + 10)} + 2$

To Find:-

• The positive Integral value of n

Solution:-

Series can be summed as:

$\sf \large 2 {}^{1} + 2 {}^{2} + 2 {}^{3} + ... + 2 {}^{n} = 2 {}^{n + 1} - 2 \\ \: \: \: \: \: \: \: \sf \large 2 {}^{2} + 2 {}^{3} + ... + 2 {}^{n} = 2 {}^{n + 1} 2 {}^{2} - 2 {}^{2} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \sf \large2 {}^{3} + ... + 2 {}^{n} = 2 {}^{n + 1} - 2 {}^{3} \\ \sf \large... \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: ... \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: ... \: \: \: \: \: \: \: \: \: \: \: \: ... \\ \sf \large \underline{ \: \: \: \: \: \: \: \: \: \: \:+ 2 {}^{n} = 2 {}^{n + 1} - 2 {}^{n} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: } \\ \sf \large \: \: \: \: \: \: \: \: \: = n(2 {}^{n + 1} ) - (2 {}^{n + 1} - 2) \\ \sf \large = 2 {}^{n + 1} (n - 1) + 2 \\ \\ \sf \large Given \: that \: 2 {}^{n + 1} (n - 1) + 2 = 2 {}^{n + 10} + 2$

$\sf \large \Rightarrow(n - 1)2 {}^{n + 1 } = 2 {}^{n + 10}$

$\sf \large \Rightarrow n - 1 = 2 {}^{9}$

$\sf \large \Rightarrow n = 2 {}^{9} + 1 = 513$

Answer:-

${ \boxed{ \sf \huge \mathfrak\green{n = 513.}}}$

Hope you have satisfied. ⚘