I am going to start step by step for every task which is asking in above question. Find the first five terms of the sequence S n = 3 n 2 + 4 n + 2 S_n = 3n^2 + 4n + 2 S n = 3 n 2 + 4 n + 2 For n = 1 , 2 , 3 , 4 , 5 n = 1, 2, 3, 4, 5 n = 1 , 2 , 3 , 4 , 5 S1= 3 ( 1 ) 2 + 4 ( 1 ) + 2 = 3 + 4 + 2 = 9 S2= 3 ( 2 ) 2 + 4 ( 2 ) + 2 = 12 + 8 + 2 = 22 S3= 3 ( 3 ) 2 + 4 ( 3 ) + 2 = 27 + 12 + 2 = 41 S4= 3 ( 3 ) 2 + 4 ( 3 ) + 2 = 27 + 12 + 2 = 41 S5=3(5)2+4(5)+2=75+20+2=97 So, the first five terms are 9 , 22 , 41 , 66 , 97 9, 22, 41, 66, 97 9 , 22 , 41 , 66 , 97 Determine the 10th term in the sequence without substituting n = 10 n = 10 directly. To find the 10th term using the difference between consecutive terms, first calculate the difference: S n + 1 − S n = ( 3 ( n + 1 ) 2 + 4 ( n + 1 ) + 2 ) − ( 3 n 2 + 4 n + 2 ) = 3 ( n 2 + 2 n + 1 ) + 4 n + 4 + 2 − 3 n 2 − 4 n − 2 = 3 n 2 + 6 n + 3 + 4 n + 4 + 2 − 3 n 2 − 4 n − 2 =6n+9 This shows that the difference between consecutive terms is 6 n + 9 6n + 9 6 n + 9 . Now, the 10th term S 10 S_{10} S 10 can be found by starting from S 1 = 9 S_1 = 9 S 1 = 9 and adding the successive differences S2= S 1 + ( 6 ( 1 ) + 9 ) = 9 + 15 = 24 S3= S 2 + ( 6 ( 2 ) + 9 ) = 24 + 21 = 45 S4= S 3 + ( 6 ( 3 ) + 9 ) = 45 + 27 = 72 S5= S 4 + ( 6 ( 4 ) + 9 ) = 72 + 33 = 105 S6= S 5 + ( 6 ( 5 ) + 9 ) = 105 + 39 = 144 S7=S6+(6(6)+9)=144+45=189 S8=S7+(6(7)+9)=189+51=240 S9=S8+(6(8)+9)=240+57=297 S10=S9+(6(9)+9)=297+63=360 So, the 10th term is S 10 = 360 S_{10} = 360 If the 20th term in the sequence equals k × 100 k \times 100 , find the value of k k . The 20th term is S 20 S_{20} : S20=3(20)2+4(20)+2 =3(400)+80+2 =1200+80+2 =1282 If S 20 = k × 100 S_{20} = k \times 100 , then 1282 = 100 k 1282 = 100k Thus, k = 1282 100 = 12.82 k = \frac{1282}{100} = 12.82 Express the sum of the first n n terms of the sequence S n S_n in the simplest algebraic form. Then, calculate the sum of the first 15 terms. The sum of the first n n terms is: Solve the equation S n = 500 S_n = 500 for n n and interpret the result. Solving: 3 n 2 + 4 n + 2 = 500 Subtract 500 from both sides: 3 n 2 + 4 n − 498 = 0 This is a quadratic equation. Solve using the quadratic formula: Interpret the root in the context of the sequence. The result gives the position n n in the sequence where the value is closest to 500, checking if it falls within the range of natural numbers.

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