Question6.045. Арифметик прогрессийн 5 - р гишүүн нь 3 - р гишүүнээс 3 - аар их ба тэдгээрийн нийлбэр нь 10 бол $2-\mathrm{p}$ гишүүнийг ол. A. 2,4 B. 4 C. 8 D. 2 E. 8,4 6.046. Арифметик прогрессийн гурав ба зургаадугаар гишүүдийн нийлбэр 3,5 . Эхний 8 гишүүний нийлбэр ол. A. 1 B. 14 C. 12 D. 15 E. 2 6.047. Арифметик прогрессийн 6 - р гишүүн нь 4 - р гишүүнээсээ 8 - аар их ба тэдгээрийн нийлбэр 33 бол $3-\mathrm{p}$ гишүүнийг ол. A. 7,5 B. 9,5 C. 8,5 D. 5 E. 5,5

Studdy Solution
Find the $2-\mathrm{p}$ term, which is the 2nd term: $$a_2 = a_1 + d$$ $$a_2 = \frac{1}{2} + \frac{3}{2}$$ $$a_2 = 2$$
The answer is $\boxed{2}$.
**Problem 6.046 Steps:**
STEP_1: Define the terms of the arithmetic progression. Let the first term be $a_1$ and the common difference be $d$.
STEP_2: The sum of the 3rd and 6th terms is 3.5: $$a_3 + a_6 = 3.5$$
STEP_3: Using the formula for the $n$-th term: $$a_3 = a_1 + 2d$$ $$a_6 = a_1 + 5d$$
Substitute these into the equation: $$a_1 + 2d + a_1 + 5d = 3.5$$ $$2a_1 + 7d = 3.5$$
STEP_4: Solve for $a_1$ and $d$. Assume $a_1 = x$ and $d = y$: $$2x + 7y = 3.5$$
Assume values or solve using additional information if available.
STEP_5: Calculate the sum of the first 8 terms: $$S_8 = \frac{8}{2} (2a_1 + 7d)$$ $$S_8 = 4 \times 3.5$$ $$S_8 = 14$$
The answer is $\boxed{14}$.
**Problem 6.047 Steps:**
STEP_1: Define the terms of the arithmetic progression. Let the first term be $a_1$ and the common difference be $d$.
STEP_2: The 6th term is 8 more than the 4th term: $$a_6 = a_4 + 8$$
The sum of the 6th and 4th terms is 33: $$a_6 + a_4 = 33$$
STEP_3: Using the formula for the $n$-th term: $$a_4 = a_1 + 3d$$ $$a_6 = a_1 + 5d$$
Substitute these into the equations: $$a_1 + 5d = a_1 + 3d + 8$$ $$a_1 + 5d + a_1 + 3d = 33$$
Simplify: $$2d = 8$$ $$2a_1 + 8d = 33$$
STEP_4: Solve for $d$: $$d = 4$$
Substitute $d$ into the second equation: $$2a_1 + 8 \times 4 = 33$$ $$2a_1 + 32 = 33$$ $$2a_1 = 1$$ $$a_1 = \frac{1}{2}$$
STEP_5: Find the $3-\mathrm{p}$ term, which is the 3rd term: $$a_3 = a_1 + 2d$$ $$a_3 = \frac{1}{2} + 2 \times 4$$ $$a_3 = \frac{1}{2} + 8$$ $$a_3 = 8.5$$
The answer is $\boxed{8.5}$.