Question & Answer

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 $S_n = 3n^2 + 4n + 2$

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

$S5=3(5)2+4(5)+2=75+20+2=97$

 Determine the 10th term in the sequence without substituting $n = 10$ directly.

To find the 10th term using the difference between consecutive terms, first calculate the difference:

Sn+1​−Sn​​=(3(n+1)2+4(n+1)+2)−(3n2+4n+2)

=3(n2+2n+1)+4n+4+2−3n2−4n−2

=3n2+6n+3+4n+4+2−3n2−4n−2

=6n+9

This shows that the difference between consecutive terms is $6n + 9$.

Now, the 10th term $S_{10}$ can be found by starting from $S_1 = 9$ and adding the successive differences

S2=S1​+(6(1)+9)=9+15=24

S3=S2​+(6(2)+9)=24+21=45

S4=S3​+(6(3)+9)=45+27=72

S5=S4​+(6(4)+9)=72+33=105

S6=S5​+(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 $S_{10} = 360$

If the 20th term in the sequence equals $k \times 100$, find the value of $k$.

The 20th term is $S_{20}$:

S20​​=3(20)2+4(20)+2

=3(400)+80+2

=1200+80+2

=1282

If $S_{20} = k \times 100$, then $1282 = 100k$

$$
Thus, $k = \frac{1282}{100} = 12.82$

Express the sum of the first $n$ terms of the sequence $S_n$in the simplest algebraic form. Then, calculate the sum of the first 15 terms.

The sum of the first $n$ terms is:

 Solve the equation $S_n = 500$ for $n$ and interpret the result.

Solving:

3n2+4n+2=500

Subtract 500 from both sides:

3n2+4n−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 $n$ in the sequence where the value is closest to 500, checking if it falls within the range of natural numbers.
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