The third term of an Arithmetic Progression (A.P.) is 23 and the sum of the first seven terms is 210. Find…

Question

The third term of an Arithmetic Progression (A.P.) is 23 and the sum of the first seven terms is 210. Find the: a. Common difference; b. First term; c. Sum of the first 20 terms.

ScanSolve AI Tutor · Accepted Answer

a) Common difference Step 1: Write down the given information using the formula for the n-th term of an A.P., a_n = a + (n−1)d, and the sum of n terms, S_n = n/2(2a + (n−1)d). Given a₃ = 23: a + (3−1)d = 23; a + 2d = 23 (1); Given S₇ = 210: 7/2(2a + (7−1)d) = 210; 7/2(2a + 6d) = 210; 7(a + 3d) = 210; a + 3d = 210/7; a + 3d = 30 (2) Step 2: Solve the system of linear equations for d. Subtract equation (1) from equation (2): (a + 3d) − (a + 2d) = 30 − 23; d = 7 The common difference is 7. b) First term Step 3: Substitute the value of d into equation (1) to find a. a + 2(7) = 23; a + 14 = 23; a = 23 − 14; a = 9 The first term is 9. c) Sum of the first 20 terms Step 4: Use the formula for the sum of n terms with n=20, a=9, and d=7. S₂₀ = 20/2(2(9) + (20−1)7); S₂₀ = 10(18 + 19 × 7); S₂₀ = 10(18 + 133); S₂₀ = 10(151); S₂₀ = 1510 The sum of the first 20 terms is 1510.

The third term of an Arithmetic Progression (A.P.) is 23 and the sum of the first seven terms is 210. Find the: a. Common difference; b. First term; c. Sum of the first 20 terms.

Answers

Step-by-step solution

Part a· 2 steps

Write down the given information using the formula for the $n$ -th term of an A.P., $a_n = a + (n-1)d$ , and the sum of $n$ terms, $S_n = \frac{n}{2}(2a + (n-1)d)$ .

Solve the system of linear equations for $d$ .

Part b· 1 step9

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

Part c· 1 step1510

Use the formula for the sum of $n$ terms with $n=20$ , $a=9$ , and $d=7$ .

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The third term of an Arithmetic Progression (A.P.) is 23 and the sum of the first seven terms is 210. Find the: a. Common difference; b. First term; c. Sum of the first 20 terms.

Answers

Step-by-step solution

Part a· 2 steps

Write down the given information using the formula for the nnn-th term of an A.P., an=a+(n−1)da_n = a + (n-1)dan​=a+(n−1)d, and the sum of nnn terms, Sn=n2(2a+(n−1)d)S_n = \frac{n}{2}(2a + (n-1)d)Sn​=2n​(2a+(n−1)d).

Solve the system of linear equations for ddd.

Part b· 1 step9

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

Part c· 1 step1510

Use the formula for the sum of nnn terms with n=20n=20n=20, a=9a=9a=9, and d=7d=7d=7.

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More Mathematics Questions

Write down the given information using the formula for the $n$ -th term of an A.P., $a_n = a + (n-1)d$ , and the sum of $n$ terms, $S_n = \frac{n}{2}(2a + (n-1)d)$ .

Solve the system of linear equations for $d$ .

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

Use the formula for the sum of $n$ terms with $n=20$ , $a=9$ , and $d=7$ .