An arithmetic progression has a first term a and common difference d. The third and seventh terms are 8 and…

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An arithmetic progression has a first term a and common difference d. The third and seventh terms are 8 and 20, respectively. (a) Find a and d. (b) Find the sum of the first 20 terms.

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Here's the solution to question 7: 7. An arithmetic progression has a first term a and common difference d. The third and seventh terms are 8 and 20, respectively. a) Find a and d. Step 1: Use the formula for the n-th term of an arithmetic progression, a_n = a + (n−1)d. For the third term (n=3): a₃ = a + (3−1)d ⇒ 8 = a + 2d (1); For the seventh term (n=7): a₇ = a + (7−1)d ⇒ 20 = a + 6d (2) Step 2: Solve the system of linear equations. Subtract equation (1) from equation (2). (a + 6d) − (a + 2d) = 20 − 8; 4d = 12; d = 12/4; d = 3 Step 3: Substitute the value of d into equation (1) to find a. 8 = a + 2(3); 8 = a + 6; a = 8 − 6; a = 2; So, a = 2 and d = 3. b) Find the sum of the first 20 terms. Step 4: Use the formula for the sum of the first n terms of an arithmetic progression, S_n = n/2(2a + (n−1)d). Here, n=20, a=2, and d=3. S₂₀ = 20/2(2(2) + (20−1)3); S₂₀ = 10(4 + (19)3); S₂₀ = 10(4 + 57); S₂₀ = 10(61); S₂₀ = 610 The sum of the first 20 terms is 610.

An arithmetic progression has a first term a and common difference d. The third and seventh terms are 8 and 20, respectively. (a) Find a and d. (b) Find the sum of the first 20 terms.

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Step-by-step solution

Part a· 3 steps

Use the formula for the $n$ -th term of an arithmetic progression, $a_n = a + (n-1)d$ .

Solve the system of linear equations.

Substitute the value of $d$ into equation (1) to find $a$ .

Part b· 1 step610

Use the formula for the sum of the first $n$ terms of an arithmetic progression, $S_n = \frac{n}{2}(2a + (n-1)d)$ .

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An arithmetic progression has a first term a and common difference d. The third and seventh terms are 8 and 20, respectively. (a) Find a and d. (b) Find the sum of the first 20 terms.

Answers

Step-by-step solution

Part a· 3 steps

Use the formula for the nnn-th term of an arithmetic progression, an=a+(n−1)da_n = a + (n-1)dan​=a+(n−1)d.

Solve the system of linear equations.

Substitute the value of ddd into equation (1) to find aaa.

Part b· 1 step610

Use the formula for the sum of the first nnn terms of an arithmetic progression, Sn=n2(2a+(n−1)d)S_n = \frac{n}{2}(2a + (n-1)d)Sn​=2n​(2a+(n−1)d).

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More Mathematics Questions

Use the formula for the $n$ -th term of an arithmetic progression, $a_n = a + (n-1)d$ .

Substitute the value of $d$ into equation (1) to find $a$ .

Use the formula for the sum of the first $n$ terms of an arithmetic progression, $S_n = \frac{n}{2}(2a + (n-1)d)$ .