If   
6
Σ
i =1
 ai  =  65, find the value of  
10
Σ
i =5
 ai −4


We are given the following information:

\[
\sum_{i=1}^{6} a_i = 65
\]

We need to find the value of:

\[
\sum_{i=5}^{10} (a_i - 4)
\]

To solve this, let's break the problem into parts.

### Step 1: Expand the given summation

\[
\sum_{i=5}^{10} (a_i - 4) = (a_5 - 4) + (a_6 - 4) + (a_7 - 4) + (a_8 - 4) + (a_9 - 4) + (a_{10} - 4)
\]

This can be rewritten as:

\[
(a_5 + a_6 + a_7 + a_8 + a_9 + a_{10}) - 6 \times 4
\]

\[
= (a_5 + a_6 + a_7 + a_8 + a_9 + a_{10}) - 24
\]

### Step 2: Use the information about the sum of \(a_i\)

Unfortunately, we do not have direct values for \(a_7, a_8, a_9, a_{10}\). However, we are told that:

\[
\sum_{i=1}^{6} a_i = 65
\]

That means:

\[
a_1 + a_2 + a_3 + a_4 + a_5 + a_6 = 65
\]

Without additional information about the specific values of \(a_7, a_8, a_9, a_{10}\), we cannot precisely determine the value of the sum. If you have further details on these variables, I can continue the solution.

Would you like to clarify or provide more information?

---

**Relative Questions:**

1. Could you provide values for \(a_7\), \(a_8\), \(a_9\), or \(a_{10}\)?
2. How does the behavior of the sequence \(a_i\) extend beyond \(a_6\)?
3. Can you specify whether \(a_i\) follows an arithmetic or geometric pattern?
4. Do you know the individual values of any of the terms \(a_5\) or \(a_6\)?
5. Is there a relationship between \(a_1\) to \(a_6\) that we can use for further insight?

**Tip:** When dealing with summations, always expand them to make the patterns and terms easier to manipulate.

If Σ(i=1 to 6) a_i = 65, find the value of Σ(i=5 to 10) (a_i - 4).

Solve the problem involving summations, where Σ(i=1 to 6) a_i = 65. Learn how to break down the problem and calculate Σ(i=5 to 10) (a_i - 4) with step-by-step reasoning, covering algebraic manipulation techniques. Suitable for high school students in Grades 9-12.

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