contestada

An arithmetic series contains 20 numbers. The first number is 102. The last number is 159. Which expression represents the sum of the series?

Respuesta :

LammettHash
If [tex]a_1=102[/tex] is the first term in the series, then the next term is [tex]a_2=a_1+k[/tex] for some fixed [tex]k[/tex], the term after that is [tex]a_3=a_2+k=a_1+2k[/tex], and so on. So the 20th term in the series would be [tex]a_{20}=a_1+(20-1)k=102+19k=159[/tex].

We have enough info to find [tex]k[/tex]:

[tex]102+19k=159\implies19k=57\implies k=3[/tex]

So the sum of the series is given by

[tex]\displaystyle\sum_{n=1}^{20}(a_1+3(n-1))=\sum_{n=1}^{20}(3n+99)=2610[/tex]

Answer: The expression is,

[tex]\frac{20}{2}(102+159)[/tex]

Step-by-step explanation:

Since, the sum of an A.P. is,

[tex]S_n=\frac{n}{2}(a+a_n)[/tex]

Where, n is the number of terms,

a is the first term of the A.P.,

[tex]a_n[/tex] is the last term of the A.P.,

Here, we have given,

a = 102, [tex]a_n[/tex] = 159, and n = 20,

Thus, the sum of the given A.P. is,

[tex]S_20=\frac{20}{2}(102+159)[/tex]

Which is the required expression.

Otras preguntas