Question

prove the sum of any number of terms of the sequence 5,7,9 starting from the first term added to 4 gives a perfect square​

Answer 1

Arithmetic Progression

Given: the given sequence is 5, 7, 9, ...

To prove: the sum of any number of terms of the sequence 5, 7, 9, ... starting from the first term added to 4 gives a perfect square

Proof:

We find the sum of the sequence for n terms..

• First term (a) = 5
• Common difference (d) = 2
• Thus the sum for n terms is
• = n/2 * [2a + (n - 1) d]
• = n/2 * [10 + 2 (n - 1)]
• = n/2 * [10 + 2n - 2]
• = n/2 * [2n + 8]
• = n (n + 4)

When 4 is added to the sum of n terms of the sequence, we get

• A = n (n + 4) + 4
• = n² + 4n + 4
• = (n + 2)², a perfect square

This completes the proof.

Conclusion: the sum of any number of terms of the sequence 5, 7, 9, ... starting from the first term added to 4 gives a perfect square.