How to calculate the first hundred terms of a sequence​

Question: what is the sum of the first 100 whole numbers?? how am i supposed to work this out efficiently? thanks

The question you asked relates back to a famous mathematician, Gauss. In elementary school in the late 1700’s, Gauss was asked to find the sum of the numbers from 1 to 100. The question was assigned as “busy work” by the teacher, but Gauss found the answer rather quickly by discovering a pattern. His observation was as follows:

1 + 2 + 3 + 4 + … + 98 + 99 + 100

Gauss noticed that if he was to split the numbers into two groups (1 to 50 and 51 to 100), he could add them together vertically to get a sum of 101.

1 + 2 + 3 + 4 + 5 + … + 48 + 49 + 50

100 + 99 + 98 + 97 + 96 + … + 53 + 52 + 51

1 + 100 = 101

2 + 99 = 101

3 + 98 = 101

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48 + 53 = 101

49 + 52 = 101

50 + 51 = 101

Gauss realized then that his final total would be 50(101) = 5050.

The sequence of numbers (1, 2, 3, … , 100) is arithmetic and when we are looking for the sum of a sequence, we call it a series. Thanks to Gauss, there is a special formula we can use to find the sum of a series:

S is the sum of the series and n is the number of terms in the series, in this case, 100.

Hope this helps!

Explanation:

Is the sequence arithmetic, does it add the same thing every time? If so take the first term and add 99 times the common difference (the number being added each time).

Example: 5, 18, 31, 44, … The common difference is 13. 5 + 99(13) = 1292

Is the sequence geometric, does it multiply by the same thing every time? If so take the first term and multiply by the common ratio (the number multiplied by each time) to the 99th power.

Example 13, 39, 117, 351, … The common ratio is 3. 13 x 3⁹⁹ = 2.333 x 10⁴⁸

transcription is a process in which information is rewritten. it is something our cells must do, in a more specialized and narrowly defined way. in biology, transcription is the process of copying out the dna sequence of a gene in the similar alphabet of rna. it needs to happen in the nucleus because that's where the dna is. dna is always inside the nucleus unless the cell is dividing.

transcription is the first step in gene expression, which involves copying a gene's dna sequence to make an rna molecule. transcription is performed by rna polymerases. it has three stages: initiation, elongation, and termination. rna molecules must be processed after transcription in eukaryotes.

explanation:

long version:

the protein synthesis consists two processes which are, transcription and translation. transcription takes place in the nucleus. it uses dna as a template to make an rna molecule. rna then leaves the nucleus and goes to a ribosome in the cytoplasm, where translation occurs. transcription is the first part of the central dogma of molecular biology: dna → rna. it is the transfer of genetic instructions in dna to messenger rna (mrna). during transcription, a strand of mrna is made that is complementary to a strand of dna.  

these three stages are:

  initiation: in which   is the beginning of transcription.occuring when the enzyme rna polymerase binds to a region of a gene called the promoter. this signals the dna to unwind so the enzyme can ‘‘read’’ the bases in one of the dna strands. the enzyme is now ready to make a strand of mrna with a complementary sequence of bases.

    elongation: in which is the addition of nucleotides to the mrna strand. rna polymerase reads the unwound dna strand and builds the mrna molecule, using complementary base pairs. there is a brief time during this process when the newly formed rna is bound to the unwound dna. during this process, an adenine (a) in the dna binds to an uracil (u) in the rna.

    termination: in which is the ending of transcription, and occurs when rna polymerase crosses a stop (termination) sequence in the gene. the mrna strand is complete, and it detaches from dna.