# Thread:

1. ## Arithmetic progression

the $N^{th}$ term of an A.P is denoted by $U_{n}$ and the sum of the $1^{st}n$ terms by $S_{N}$ in a certain A.P ;; $U_{5}+U_{16}=44$ and $S_{18}=3S_{10}.$calculate the value of the first term and the common difference.

i have been trying to get two equations from the given equation using the rules for arithmetic progression but i have been unable to do so..

i really need some help coming up with these equations.

2. Originally Posted by sigma1
the $N^{th}$ term of an A.P is denoted by $U_{n}$ and the sum of the $1^{st}n$ terms by $S_{N}$ in a certain A.P ;; $U_{5}+U_{16}=44$ and $S_{18}=3S_{10}.$calculate the value of the first term and the common difference.

i have been trying to get two equations from the given equation using the rules for arithmetic progression but i have been unable to do so..

i really need some help coming up with these equations.

Using the formula or the n-th term in an A.P., for all $n\,,\,\,u_n=u_1+(n-1)d\Longrightarrow 44=u_5+u_{16}=$ $u_1+4d+u_1+15d=2u_1+19d\Longrightarrow (I)\,\,2u_1+19d=44$ , and using the formula for the sum of consecutive elements in an A.P. we get

$3S_{10}=3\cdot \frac{10}{2}\left(2u_1+9d\right)=\frac{18}{2}\left (2u_1+17d\right)=S_{18}$ $\Longrightarrow 10u_1+45d=6u_1+51d\Longrightarrow (II)\,\,2u_1-3d=0$ , and there you have two linear eq's in two unknowns: $u_1\,\,\,and\,\,\,d$

Tonio
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