https://file.aiquickdraw.com/mathbot/file/1725515656912lbjt6rg6.jpg

Let me transcribe and solve the mathematical problems from the image. Here is the text transcription:

1. (i) কোন ধারার ধারা \(n^2\) এবং \(n \neq 1\) এর যোগফল \(n^2 + n^2 = n^2\)।
   
   (ii) একটি জ্যামিতিক ধারার ৪র্থ পদ \(\frac{1}{3}\) এবং ৯ম পদ \(\frac{\sqrt{3}}{81}\)।
   
2. \( \sqrt{5}/5 \) এর 5 তম ঘাত কত?

3. (i)  \(m \neq n\) হলে \(m + n\) এর পদ এবং \(m + n\) সংখ্যক পদে সন্তানের নির্ণয় কর।

4. (ii) \(n\) থেকে ধারাটি নির্ণয় কর।

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Let me break down the individual questions and provide solutions.

### Question 1 (i):

This part asks for the sum of some series involving \(n^2\). However, the transcription is unclear. Could you kindly confirm if the series provided was geometric or arithmetic, or if any other specifics are there?

### Question 1 (ii):

We are given:
- 4th term \(T_4 = \frac{1}{3}\)
- 9th term \(T_9 = \frac{\sqrt{3}}{81}\)

This is a geometric series. The general form of the nth term of a geometric sequence is:
\[ T_n = ar^{n-1} \]

For \(T_4\) and \(T_9\), we have the following two equations:
\[ ar^3 = \frac{1}{3} \]
\[ ar^8 = \frac{\sqrt{3}}{81} \]

We can solve for the common ratio \(r\) and the first term \(a\).

### Question 2:

The expression \(\sqrt{5}/5\) raised to the 5th power is:
\[
\left( \frac{\sqrt{5}}{5} \right)^5
\]

This can be simplified to find the result.

### Question 3 (i):

This question involves the sum of an arithmetic series with \(m + n\) terms. Additional clarification is needed, particularly on the "সন্তানের নির্ণয় কর" part.

### Question 4:

The question asks to determine the series from \(n\). It seems to request the formula or type of series based on the value of \(n\), but more details would be useful.

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I'll now proceed with solving the clear parts of the questions. Let's first focus on solving the geometric series from 1(ii) and the expression from question 2.

1. (i) কোন ধারার ধারা n^2 এবং n ≠ 1 এর যোগফল n^2 + n^2 = n^2। (ii) একটি জ্যামিতিক ধারার ৪র্থ পদ 1/3 এবং ৯ম পদ √3/81। 2. √5/5 এর 5 তম ঘাত কত? 3. (i) m ≠ n হলে (m + n) এর পদ এবং (m + n) সংখ্যক পদে সন্তানের নির্ণয় কর। 4. (ii) n থেকে ধারাটি নির্ণয় কর।

Learn how to solve problems related to arithmetic and geometric series. Understand the sum of square numbers, calculate terms in a geometric sequence, and work with exponents, as seen in the problem with √5/5 raised to the 5th power.

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