Questions

• Arithmetic and Geometric Sequences and Their Summations Understand the concept and the properties of geometric sequences Understand the general term of a geometric sequence

  給定$T(n)$是首項為$a$且公比為$r$的等比數列的第$n$項，其中$a>0$且$0<r<1$。設$x_{1}, y_{1}$和$z_{1}$分別為數組$\{T(1), T(2), T(3), \ldots, T(49)\}$的平均數、四分位距和方差，而$x_{2}, y_{2}$和$z_{2}$分別為數組$\{T(51), T(52), T(53), \ldots, T(99)\}$的平均數、四分位距和方差。以下哪項必定為真？ I. $x_{1}=0.02 x_{2}$ II. $y_{2}=r^{50} y_{1}$ III. $z_{1}=\frac{1}{r^{100}} z_{2}$

  2024

• Arithmetic and Geometric Sequences and Their Summations Explore the general formulae of the sum to infinity for certain geometric sequences and use the formulae to solve related problems

  給定一個等比數列的無窮和為12，且數列的第二項為3，求公比。

  2024

• Arithmetic and Geometric Sequences and Their Summations Explore the general formulae of the sum to infinity for certain geometric sequences and use the formulae to solve related problems

  給定一個等比數列的無窮和為20，公比為0.5，求該數列的第四項。

  2024

• Arithmetic and Geometric Sequences and Their Summations Understand the general term of an arithmetic sequence

  給定等差數列：5, 8, 11, 14, ...，確定序列中的哪一項是50。A. 15。B. 16。C. 17。D. 它不是序列的一部分。正確答案：A解釋：等差數列的通項公式為：\[ a_n = a_1 + (n-1)d \]其中 \( a_1 \) 是第一項，\( d \) 是公差。對於給定的序列：\[ a_1 = 5 \]\[ d = 8 - 5 = 3 \]我們需要確定50是否是序列中的一項。因此，我們設 \( a_n = 50 \) 並求解 \( n \)：\[ 50 = 5 + (n-1)3 \]\[ 50 = 5 + 3n - 3 \]\[ 50 = 3n + 2 \]\[ 48 = 3n \]\[ n = \frac{48}{3} \]\[ n = 16 \]

  2024

• Arithmetic and Geometric Sequences and Their Summations Understand the concept and the properties of arithmetic sequences

  一家公司的利潤\( P \)與銷售單位數\( u \)和每單位價格\( p \)成正比，並與廣告成本\( A \)成部分比。如果當銷售100單位，每單位售價為\$20，廣告成本為\$2000時，利潤為\$5000，那麼當銷售150單位，每單位售價為\$25，廣告成本為\$3000時，利潤將是多少？

  2024

• Arithmetic and Geometric Sequences and Their Summations Understand the concept and the properties of arithmetic sequences

  考慮一個等差序列，其中第一項\( T_1 \)是5，公差\( d \)是3。如果\( T_n \)代表序列的第n項，找出\( T_{10} \)的值。另外，驗證由\( 2T_1 + 1, 2T_2 + 1, 2T_3 + 1, \ldots \)形成的序列是否也是一個等差序列，並確定其公差。

  2024

• Arithmetic and Geometric Sequences and Their Summations Understand the concept and the properties of arithmetic sequences

  考慮一個等差序列，其中第一項 \( a_1 \) 為 3，公差 \( d \) 為 2。設 \( S_n \) 代表這個序列前 \( n \) 項的和。求最小的整數 \( n \)，使得 \( \log_2(S_n) > 5 \)。

  2024

• Arithmetic and Geometric Sequences and Their Summations Understand the concept and the properties of arithmetic sequences

  考慮一個等差數列，其中第一項是未知的，記為 \( a \)，公差也是未知的，記為 \( d \)。該數列的第5項是20，第10項是35。找出第一項 \( a \) 的值。

  2024

• Arithmetic and Geometric Sequences and Their Summations Understand the concept and the properties of arithmetic sequences

  考慮一個等差數列，其中第一項\(a_1\)是3，公差\(d\)是2。設\(a_n\)為該數列的第\(n\)項。如果\(a_n\)乘以\(2^n\)，新數列的前5項和是多少？

  2024

• Arithmetic and Geometric Sequences and Their Summations Understand the general term of an arithmetic sequence

  給定等差數列：3, 7, 11, 15, ...，找出通項\(a_n\)。

  2024