In a number pattern, it is given that 17 + 17 = 2895, 18 + 18 = 3245 and 19 + 19 = 3615. Following the same pattern, what should be the value of 23 + 23?

Introduction / Context:
Here we are not dealing with ordinary arithmetic addition. Instead, the expression "a + a" is being used to define a number pattern. The left hand side gives a clue for generating a new number on the right hand side. We must detect the hidden rule from the given examples and then apply that rule to the new pair 23 + 23.

Given Data / Assumptions:

• 17 + 17 is mapped to 2895.
• 18 + 18 is mapped to 3245.
• 19 + 19 is mapped to 3615.
• We assume the same rule applies consistently to 23 + 23.
• Usual algebraic equality is not intended; this is a pattern or coding problem.

Concept / Approach:
We look for a relationship between the given number a and the resulting digits on the right. One immediate observation is that 17^2 = 289, 18^2 = 324 and 19^2 = 361. These three squares form the first three digits of each four digit result. The last digit is 5 in every case. So the pattern appears to be: for a + a, write a^2 and then append the digit 5 at the end to form the final number.

Step-by-Step Solution:
Step 1: Check the first example. For 17 + 17, compute 17^2 = 289. Step 2: Compare with 2895. The first three digits 289 match the square, and an extra 5 appears at the end. Step 3: Check the second example. For 18 + 18, 18^2 = 324, and the result is 3245, again square followed by 5. Step 4: Check the third example. For 19 + 19, 19^2 = 361, and the result is 3615, again square followed by 5. Step 5: Having confirmed a stable pattern, apply it to 23 + 23. Step 6: Compute 23^2 = 529. Step 7: Append digit 5 to obtain 5295.

Verification / Alternative check:
Another way to check is to verify that no simpler interpretation, such as mixing ordinary addition or concatenation of sums and differences, fits all three examples so neatly. The square plus appended digit 5 pattern matches every case exactly, which makes it the most consistent rule. Therefore using it for 23 is justified.

Why Other Options Are Wrong:
4005, 5765, 2565 and 4845 do not follow the rule "square of the number followed by 5". For example, 23^2 is 529, not 400, 576, 256 or 484. These options either square some other number or mix unrelated manipulations, so they do not represent the discovered pattern.

Common Pitfalls:
Pattern questions often tempt students to try arbitrary combinations such as a^2 + a, a^2 - a, or appending sums of digits. Without checking all given pairs, such guesses can lead to the wrong rule. The reliable method is to test any hypothesis across each example, and only accept a pattern that works uniformly for all of them.

Final Answer:
According to the rule "write the square of the number and then append 5", the value corresponding to 23 + 23 is 5295.