Respuesta :

LammettHash

Consecutive terms differ by 1.5, so this is an arithmetic sequence given by

[tex]\begin{cases}a_1=19\\a_n=a_{n-1}+1.5&\text{for }n>1\end{cases}[/tex]

So we have

[tex]a_2=a_1+1.5[/tex]

[tex]a_3=a_2+1.5=a_1+2\cdot1.5[/tex]

[tex]a_4=a_3+1.5=a_1+3\cdot1.5[/tex]

and so on, up to

[tex]a_n=a_1+1.5(n-1)=17.5+1.5n[/tex]

The sum of the first [tex]N[/tex] terms is 10,900:

[tex]\displaystyle\sum_{n=1}^Na_n=\sum_{n=1}^N(17.5+1.5n)=17.5N+1.5\dfrac{N(N+1)}2=10,900[/tex]

[tex]\implies17.5N+0.75N(N+1)=10,900[/tex]

[tex]\implies0.75N^2+18.25N-10,900=0[/tex]

[tex]\implies N=109[/tex] (we ignore the negative solution)

so there are 109 terms in the series.

gmany

Answer:

109

Step-by-step explanation:

[tex]19,\ 20.5,\ 22,\ 23.5,\ ...,\ 181\\\\\text{It's the arithmetic sequence with:}\\\\a_1=19,\ a_n=181.\\\\\text{The sum of n terms of this sequence is equal to 10,900.}\\\\\text{the formula of a sum of n terms of an arithmetic sequence:}\\\\S_n=\dfrac{a_1+a_n}{2}\cdot n\\\\\text{Substitute:}\ S_n=10,900,\ a_1=19,\ a_n=181:\\\\\dfrac{19+181}{2}\cdot n=10,900\\\\\dfrac{200}{2}n=10,900\\\\100n=10,900\qquad\text{divide both sides by 100}\\\\n=109[/tex]

Otras preguntas