For the following two (02) items : Let $(6+10+14 + ... \text{up to } m \text{ terms})$ $=(1+3+5+7+ ... \text{up to } n \text{ terms})$ where $m < 25$ and $n < 25$.
Analyzing the Arithmetic Series
The question asks us to find the number of possible values for '\(m\)' given an equality between the sums of two arithmetic progressions (APs), subject to constraints on '\(m\)' and '\(n\)' (\(m < 25\) and \(n < 25\)).
Let's first find the formula for the sum of each series:
-
First Series: \(6, 10, 14, ...\) up to \(m\) terms.
- First term (\(a_1\)) = 6
- Common difference (\(d_1\)) = \(10 - 6 = 4\)
- The sum of the first \(m\) terms (\(S_m\)) is given by the formula: \(S_m = \frac{m}{2} [2a_1 + (m-1)d_1]\)
- Substituting the values: \(S_m = \frac{m}{2} [2(6) + (m-1)4]\) \(S_m = \frac{m}{2} [12 + 4m - 4]\) \(S_m = \frac{m}{2} [8 + 4m]\) \(S_m = m(4 + 2m)\) \(S_m = 2m^2 + 4m\)
-
Second Series: \(1, 3, 5, 7, ...\) up to \(n\) terms.
- First term (\(a_2\)) = 1
- Common difference (\(d_2\)) = \(3 - 1 = 2\)
- The sum of the first \(n\) terms (\(S_n\)) is given by the formula: \(S_n = \frac{n}{2} [2a_2 + (n-1)d_2]\)
- Substituting the values: \(S_n = \frac{n}{2} [2(1) + (n-1)2]\) \(S_n = \frac{n}{2} [2 + 2n - 2]\) \(S_n = \frac{n}{2} [2n]\) \(S_n = n^2\)
Setting up the Equality Equation
The problem states that the sums are equal:
\(S_m = S_n\) \(2m^2 + 4m = n^2\)Finding Possible Values of m
We are given the constraints \(m < 25\) (meaning \(1 \le m \le 24\)) and \(n < 25\) (meaning \(1 \le n \le 24\)). We need to find integer values of \(m\) within its range that result in an integer value of \(n\) within its range, satisfying the equation \(n^2 = 2m^2 + 4m\).
Let's test values of \(m\) from 1 to 24:
| m | Calculation for \(n^2 = 2m^2 + 4m\) | \(n^2\) | Is \(n^2\) a perfect square? | n | Is \(n < 25\)? | Valid Pair (m, n)? |
|---|---|---|---|---|---|---|
| 1 | \(2(1)^2 + 4(1)\) | 6 | No | - | - | No |
| 2 | \(2(2)^2 + 4(2)\) | 16 | Yes | 4 | Yes | Yes |
| 3 | \(2(3)^2 + 4(3)\) | 30 | No | - | - | No |
| 4 | \(2(4)^2 + 4(4)\) | 48 | No | - | - | No |
| 5 | \(2(5)^2 + 4(5)\) | 70 | No | - | - | No |
| ... | ... | ... | ... | ... | ... | ... |
| 15 | \(2(15)^2 + 4(15)\) | 510 | No | - | - | No |
| 16 | \(2(16)^2 + 4(16)\) | 576 | Yes | 24 | Yes | Yes |
| 17 | \(2(17)^2 + 4(17)\) | 646 | No | - | - | No |
| ... | ... | ... | ... | ... | ... | ... |
| 24 | \(2(24)^2 + 4(24)\) | 1248 | No | - | - | No |
By testing values, we find two pairs \((m, n)\) that satisfy the equation and the constraints:
- When \(m=2\), \(n^2 = 2(2^2) + 4(2) = 8 + 8 = 16\), so \(n=4\). Both \(m=2\) and \(n=4\) are less than 25.
- When \(m=16\), \(n^2 = 2(16^2) + 4(16) = 2(256) + 64 = 512 + 64 = 576\), so \(n=24\). Both \(m=16\) and \(n=24\) are less than 25.
For any other value of \(m\) between 1 and 24, \(n^2 = 2m^2 + 4m\) does not result in a perfect square for \(n\), or the resulting \(n\) is 25 or greater.
Conclusion on Possible Values of m
The possible values for \(m\) are 2 and 16. Therefore, there are exactly two possible values for \(m\).
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