Question

Express the following as the sum of two consecutive integers
(i) 11²
(ii) 15²
(iii) 19²
(iv) 29²​

Answer 1

Answer

To Do

• To express the given numbers as the sum of two consecutive integers -

1. ( 11 )²
2. ( 15 )²
3. ( 19 )²
4. ( 29 )²

Step By Step Explanation

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1. ( 11 )² => 121

Let us consider the two consecutive integers be x and x + 1.

Now,

$\longmapsto \sf121 = x + x + 1 \\ \\\longmapsto \sf 121 = 2x + 1 \\ \\\longmapsto \sf 121 - 1 = 2x \\ \\\longmapsto \sf 120 = 2x \\ \\\longmapsto \sf \cancel\cfrac{120}{2} = x \\ \\\longmapsto \sf 60 = x$

Therefore two consecutive integers are 60 and 61.

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2. ( 15 )² => 225

Let us consider the two consecutive integers be y and y + 1.

Now,

$\longmapsto \sf225 = y + y + 1 \\ \\ \longmapsto \sf225 = 2y + 1 \\ \\ \longmapsto \sf225 - 1 = 2y \\ \\ \longmapsto \sf224 = 2y \\ \\ \longmapsto \sf \cancel\cfrac{224}{2} = y \\ \\ \longmapsto \sf 112 = y$

Therefore two consecutive integers are 112 and 113.

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3. ( 19 )² => 361

Let us consider the two consecutive integers be z and z + 1.

Now,

$\longmapsto \sf361 = z + z + 1 \\ \\ \longmapsto \sf361 = 2z + 1 \\ \\ \longmapsto \sf361 - 1 = 2z \\ \\ \longmapsto \sf360 = 2z \\ \\ \longmapsto \sf \cancel\cfrac{360}{2} = z \\ \\ \longmapsto \sf180 = z$

Therefore two consecutive integers are 180 and 181.

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4. ( 29 )² => 841

Let us consider the two consecutive integers be w and w + 1.

Now,

$\longmapsto \sf841 = w + w + 1 \\ \\\longmapsto \sf 841 = 2w + 1 \\ \\ \longmapsto \sf841 - 1 = 2w \\ \\\longmapsto \sf 840 = 2w \\ \\ \longmapsto \sf \cancel \cfrac{840}{2} = w \\ \\\longmapsto \sf 420 = w$

Therefore two consecutive integers are 420 and 421.

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