beninar6u

2022-10-09

Determine whether the sequence ln1,ln2,ln3,ln4,ln5,... is arithmetic and if it is, what is the common difference

Derick Ortiz

Beginner2022-10-10Added 11 answers

The term of an AP sequence takes the form:

$\{a,a+d,a+2d,a+3d,,...\}$

Where:

a=first term

d=common difference

If the given sequence:

$\{\mathrm{ln}1,\mathrm{ln}2,\mathrm{ln}3,...\}$

were an AP the clearly we have first term given by:

$a=\mathrm{ln}1(=0)$

And the second term would satisfy:

$a+d=\mathrm{ln}2$

$\Rightarrow \mathrm{ln}1+d=\mathrm{ln}2$

$\Rightarrow d=\mathrm{ln}2-\mathrm{ln}1=\mathrm{ln}2$

Thus, the third term would be given by:

$a+2d=\mathrm{ln}1+2\cdot \mathrm{ln}2=\mathrm{ln}4\ne \mathrm{ln}3$

Hence, the terms are not part of an AP

$\{a,a+d,a+2d,a+3d,,...\}$

Where:

a=first term

d=common difference

If the given sequence:

$\{\mathrm{ln}1,\mathrm{ln}2,\mathrm{ln}3,...\}$

were an AP the clearly we have first term given by:

$a=\mathrm{ln}1(=0)$

And the second term would satisfy:

$a+d=\mathrm{ln}2$

$\Rightarrow \mathrm{ln}1+d=\mathrm{ln}2$

$\Rightarrow d=\mathrm{ln}2-\mathrm{ln}1=\mathrm{ln}2$

Thus, the third term would be given by:

$a+2d=\mathrm{ln}1+2\cdot \mathrm{ln}2=\mathrm{ln}4\ne \mathrm{ln}3$

Hence, the terms are not part of an AP

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