Bank Soal Matematika SMA Notasi Sigma

Perhatikan beberapa sifat notasi sigma berikut!

1. $\sum_{i=1}^nU_i=U_1+U_2+U_3+\dots+U_n$
2. $\sum_{i=1}^nU_i=\sum_{j=1}^nU_j$
3. $\sum_{i=1}^nC=Cn$
4. $\sum_{i=1}^nCU_i=C\sum_{i=1}^nU_i$
5. $\sum_{i=1}^n\left(U_i\pm V_i\right)=\sum_{i=1}^nU_i\pm\sum_{i=1}^nV_i$
6. $\sum_{i=1}^n\left(U_i+V_i\right)^2=\sum_{i=1}^nU_i^2+2\sum_{i=1}^nU_iV_i+\sum_{i=1}^nV_i^2$
7. $\sum_{i=1}^nU_i+\sum_{i=n+1}^mU_i=\sum_{i=1}^mU_i$
8. $\sum_{i=1}^nU_i=\sum_{i=0}^{n-1}U_{i+1}=\sum_{i=2}^{n+1}U_{i-1}$

dengan

$n$ dan $m$: suatu bilangan bulat non-negatif

$U_i$: rumus suku ke-$i$ suatu deret

$V_i$: rumus suku ke-$i$ suatu deret

$C$: suatu konstanta

Diketahui notasi sigma berikut

$5\sum_{n=1}^8\left(n+2\right)-3\sum_{n=1}^8\left(n-1\right)$

yang sama dengan $\sum_{n=1}^8f\left(n\right)$.

Akan dicari nilai $f\left(n\right)$ yang tepat

Berdasarkan sifat 4 diperoleh

$5\sum_{n=1}^8\left(n+2\right)-3\sum_{n=1}^8\left(n-1\right)=\sum_{n=1}^85\left(n+2\right)+\sum_{n=1}^8\left(-3\right)\left(n-1\right)$

$5\sum_{n=1}^8\left(n+2\right)-3\sum_{n=1}^8\left(n-1\right)=\sum_{n=1}^8\left(5n+10\right)+\sum_{n=1}^8\left(-3n+3\right)$

Berdasarkan sifat 5 diperoleh

$5\sum_{n=1}^8\left(n+2\right)-3\sum_{n=1}^8\left(n-1\right)=\sum_{n=1}^8\left(\left(5n+10\right)+\left(-3n+3\right)\right)$

$5\sum_{n=1}^8\left(n+2\right)-3\sum_{n=1}^8\left(n-1\right)=\sum_{n=1}^8\left(5n+10-3n+3\right)$

$5\sum_{n=1}^8\left(n+2\right)-3\sum_{n=1}^8\left(n-1\right)=\sum_{n=1}^8\left(5n-3n+10+3\right)$

$5\sum_{n=1}^8\left(n+2\right)-3\sum_{n=1}^8\left(n-1\right)=\sum_{n=1}^8\left(2n+13\right)$

Jadi nilai yang tepat adalah

$f\left(n\right)=2n+13$