Ajụjụ Ihe Nlereanya Na-ekwu Maka Usoro Geometric
Usoro geometric bụ echiche dị mkpa na mgbakọ na mwepụ nke a na-akụzikarị n'ụlọ akwụkwọ sekọndrị. Usoro ndị a nwere otu nọmba, nke ọ bụla n'ime ha bụ ngwaahịa nke ọnụọgụgụ gara aga site na ihe a na-akpọ "ratio." Isiokwu a ga-ekpuchi ọtụtụ ihe atụ nke nsogbu na mkparịta ụka nke usoro geometric, na-atụ anya inyere ndị na-agụ akwụkwọ aka ịghọta echiche a nke ọma.
Ịghọta Usoro Geometric
Usoro geometry bụ usoro ọnụọgụgụ e ji eme ọnụọgụgụ site n'ịba ụba ọnụọgụgụ mbụ (a) site na oke edobere (r). N'ozuzu, ụdị izugbe nke usoro geometry bụ:
\[a, ar, ar^2, ar^3, \ldots, ar^{n-1} \]
Ebe a:
- "a" bụ okwu mbụ nke usoro iheomume ahụ.
– “r” bụ oke nke otu okwu na okwu gara aga.
– “n” bụ okwu nke n’usoro.
Ajụjụ na Mkparịta ụka Ihe Nlereanya
Ka anyị tụlee ụfọdụ nsogbu ihe atụ iji ghọtakwuo gbasara usoro geometric.
Ajụjụ Ihe Nlereanya nke 1
Ajụjụ:
Ọ bụrụ na e jiri usoro geometric mee ihe, nke okwu mbụ (a) bụ 3, nke (r) bụ 2. Kpebie:
1. Okwu nke iri nke usoro ahụ.
2. Ngụkọta nke okwu isii mbụ nke usoro ahụ.
Azịza:
1. Enwere ike ịgbakọ okwu nke ise (U5) site na iji usoro maka okwu nke n'ime usoro geometric, ya bụ:
\[ U_n = a \cdot r^{n-1} \]
Iji a = 3, r = 2, na n = 5 dochie usoro ahụ:
\[ U_5 = 3 \cdot 2^{5-1} \]
\[ U_5 = 3 \cdot 2^4 \]
\[ U_5 = 3 \cdot 16 \]
\[ U_5 = 48 \]
Ya mere, okwu nke iri na abụọ nke usoro ahụ bụ 48.
2. Enwere ike ịgbakọ mkpokọta nke okwu isii mbụ (S6) nke usoro geometric site na iji usoro maka mkpokọta nke okwu n mbụ, ya bụ:
\[ S_n = a \left( \frac{r^n – 1}{r – 1} \nri) \]
Iji a = 3, r = 2, na n = 6 dochie usoro ahụ:
\[ S_6 = 3 \aka ekpe( \frac{2^6 – 1}{2 – 1} \aka nri) \]
\[ S_6 = 3 \aka ekpe( \frac{64 – 1}{1} \aka nri) \]
\[ S_6 = 3 \aka ekpe( 63 \aka nri) \]
\[ S_6 = 189 \]
Ya mere, ngụkọta nke okwu isii mbụ nke usoro ahụ bụ 189.
Ajụjụ Ihe Nlereanya nke 2
Ajụjụ:
Usoro geometric nwere okwu nke atọ nke 27 na okwu nke ise nke 243. Chọpụta uru nke okwu mbụ (a) na oke (r).
Azịza:
E nyere U3 = 27 na U5 = 243. Iji usoro maka okwu nke n'ime usoro geometric:
\[ U_n = a \cdot r^{n-1} \]
Maka U3:
\[ U_3 = a \cdot r^2 \]
\[ 27 = a \cdot r^2 \] \[ (1) \]
Maka U5:
\[ U_5 = a \cdot r^4 \]
\[ 243 = a \cdot r^4 \] \[ (2) \]
Iji nha nhata (1) na (2) tụnyere iji wepụ a:
\[ \frac{U_5}{U_3} = \frac{a \cdot r^4}{a \cdot r^2} \]
\[ \frac{243}{27} = r^2 \]
\[ 9 = r^2 \]
\[ r = 3 \ederede{ ma ọ bụ } r = -3 \]
Tinye uru nke r n'ime nha nhata (1):
Ọ bụrụ na \( r = 3 \):
\[ 27 = a \cdot 3^2 \]
\[ 27 = a \cdot 9 \]
\[a = 3 \]
Ọ bụrụ na \( r = -3 \):
\[ 27 = a \cdot (-3)^2 \]
\[ 27 = a \cdot 9 \]
\[a = 3 \]
Ya mere, okwu mbụ (a) bụ 3, oke (r) nwekwara ike ịbụ 3 ma ọ bụ -3.
Ajụjụ Ihe Nlereanya nke 3
Ajụjụ:
Chọta ngụkọta enweghị ngwụcha nke usoro geometric ndị a ma ọ bụrụ na okwu mbụ (a) bụ 8 na oke (r) bụ 1/2.
Azịza:
Enwere ike ịgbakọ ngụkọta na-enweghị njedebe nke usoro geometric site na iji usoro a:
\[ S_{\infty} = \frac{a}{1 – r} \]
Iji a = 8 na r = 1/2 dochie usoro a:
\[ S_{\infty} = \frac{8}{1 – \frac{1}{2}} \]
\[ S_{\infty} = \frac{8}{\frac{1}{2}} \]
\[ S_{\infty} = ugboro asatọ abụọ
\[ S_{\infty} = 16 \]
Ya mere, ngụkọta enweghị ngwụcha nke usoro geometric bụ 16.
Ajụjụ Ihe Nlereanya nke 4
Ajụjụ:
Usoro geometric nwere okwu nke abụọ nke 12 na okwu nke anọ nke 108. Chọpụta oke na okwu mbụ nke usoro ahụ.
Azịza:
E nyere \( U_2 = 12 \) na \( U_4 = 108 \). Iji usoro maka okwu nke n nke usoro geometric:
Maka \( U_2 \):
\[ U_2 = a \cdot r \]
\[ 12 = a \cdot r \] \[ (1) \]
Maka \( U_4 \):
\[ U_4 = a \cdot r^3 \]
\[ 108 = a \cdot r^3 \] \[ (2) \]
Iji nha nhata (1) na (2) tụnyere iji wepụ a:
\[ \frac{U_4}{U_2} = \frac{a \cdot r^3}{a \cdot r} \]
\[ \frac{108}{12} = r^2 \]
\[ 9 = r^2 \]
\[ r = 3 \ederede{ ma ọ bụ } r = -3 \]
Tinye uru nke r n'ime nha nhata (1):
Ọ bụrụ na \( r = 3 \):
\[ 12 = a \cdot 3 \]
\[a = 4 \]
Ọ bụrụ na \( r = -3 \):
\[ 12 = a \cdot (-3) \]
\[a = -4 \]
Ya mere, okwu mbụ (a) nwere ike ịbụ 4 ma ọ bụ -4, oke (r) nwekwara ike ịbụ 3 ma ọ bụ -3.
Mmechi
Usoro geometric bụ echiche mgbakọ na mwepụ dị mkpa a na-ejikarị eme ihe n'ọtụtụ ngalaba dị iche iche. Site n'ịghọta ihe ndị bụ isi na ime nkà idozi nsogbu, anyị nwere ike ịbụ ndị ọkachamara n'ịghọta na itinye echiche ahụ n'ọrụ. Isiokwu a gụnyere ọtụtụ ihe atụ nke nsogbu na mkparịta ụka iji nyere ndị na-agụ akwụkwọ aka ịmụta ma ghọta usoro geometric nke ọma. Anyị nwere olileanya na nke a ga-abara uru!
