Solve the following mathematical problems:

Prove that the sum of the odd numbers from 1 to 125 inclusive is equal to the sum of the odd numbers from 169 to 209 inclusive.

Created: 3 months ago | Updated: 3 months ago

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The sum of the odd numbers from 1 to 125
Inclusive = 1+3+5+...……+125
It's an arithmetic progression

Where, a = 1; d=3-1=2

$L e t, n^{t h} t e r m = 125 a + (n - 1) d = 125 \Rightarrow 1 + (n - 1) 2 = 125 \Rightarrow 2 n - 2 = 125 - 1 \Rightarrow 2 n = 124 + 2 ∴ n = \frac{126}{2} = 63 S n = \frac{n}{2} (2 a + (n - 1) d} = \frac{63}{2} {2 x 1 + (63 - 1) \times 2) = \frac{63}{2} (2 + 124) = \frac{63}{2} \times 126 = 3969$

Again, sum of the add numbers form 169 to 209 inclusive = 169 + 171 + 173 +.......+ 209

Here, a = 169

d = 171-169 = 2

$L e t, n^{t h} t e r m = 209 \Rightarrow a + (n - 1) d = 209 \Rightarrow 169 + (n - 1) 2 = 209 \Rightarrow 2 n - 2 = 209 - 169 \Rightarrow 2 n = 40 + 2 ∴ n = \frac{42}{2} = 21 s n = \frac{n}{2} \{2 a + (n - 1) d\} = \frac{21}{2} \{2 \times 169 + (21 - 1) \times 2\} = \frac{21}{2} (338 + 40) = \frac{21}{2} \times 378 = 3969 1 + 3 + 5 . . . . . . . . . + 125 = 169 + 171 + 173 + . . . . . . + 209 [p r o v e d]$

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Solve the following mathematical problems:

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