Arithmetic Series (Sum)
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Question 1 of 5
1. Question
Find the sum of the sequence`6+11+16+…+236`- `S_n=` (5687)
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Sum of an Arithmetic Sequence
$$S_{\color{#9a00c7}{n}}=\frac{\color{#9a00c7}{n}}{2}[\color{#e65021}{a}+\color{#00880A}{l}]$$Common Difference Formula
$$d=U_2-U_1=U_3-U_2$$General Rule of an Arithmetic Sequence
$$U_{\color{#9a00c7}{n}}=\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$First, solve for the value of `d`.$$\color{#00880A}{d}$$ `=` $$U_2-U_1$$ `=` $$11-6$$ Substitute the first and second term `=` `5` Next, substitute the known values to the general rule and solve for `n``\text(Nth term)``[U_n]` `=` `236` `\text(First term)``[a]` `=` `6` `\text(Common Difference)``[d]` `=` `5` $$U_{\color{#9a00c7}{n}}$$ `=` $$\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$ $$236$$ `=` $$\color{#e65021}{6}+[(\color{#9a00c7}{n}-1)\color{#00880A}{5}]$$ Substitute known values `236` `=` `6+5n-5` Distribute `236` `-1` `=` `1+5n` `-1` Subtract `1` from both sides `235``divide5` `=` `5n``divide5` Divide both sides by `5` `47` `=` `n` `n` `=` `47` Finally, substitute the known values to the sum formula`\text(Number of terms) [n]` `=` `47` `\text(First Term) [a]` `=` `6` `\text(Last Term) [l]` `=` `236` $$S_{\color{#9a00c7}{n}}$$ `=` $$\frac{\color{#9a00c7}{n}}{2}[\color{#e65021}{a}+\color{#00880A}{l}]$$ $$S_{\color{#9a00c7}{47}}$$ `=` $$\frac{\color{#9a00c7}{47}}{2}[\color{#e65021}{6}+\color{#00880A}{236}]$$ Substitute known values `=` $$\frac{47}{2}[242]$$ Evaluate `=` $$\frac{11374}{2}$$ `=` `5687` `S_n=5687` -
Question 2 of 5
2. Question
Find the sum of the first `20` terms given that:`U_1=8``U_18=76`- `S_20=` (920)
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Sum of an Arithmetic Sequence
$$S_{\color{#9a00c7}{n}}=\frac{\color{#9a00c7}{n}}{2}[2\color{#e65021}{a}+(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$General Rule of an Arithmetic Sequence
$$U_{\color{#9a00c7}{n}}=\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$First, substitute the known values to the general rule and solve for `d``\text(18th term)``[U_18]` `=` `76` `\text(First term)``[a]` `=` `8` $$U_{\color{#9a00c7}{n}}$$ `=` $$\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$ $$U_{\color{#9a00c7}{18}}$$ `=` $$\color{#e65021}{8}+[(\color{#9a00c7}{18}-1)\color{#00880A}{d}]$$ Substitute known values `76` `-8` `=` `8+17d` `-8` Subtract `8` from both sides `68``divide17` `=` `17d``divide17` Divide both sides by `17` `4` `=` `d` `d` `=` `4` Finally, substitute the known values to the sum formula`\text(Number of terms) [n]` `=` `20` `\text(First Term) [a]` `=` `8` `\text(Common Difference) [d]` `=` `4` $$S_{\color{#9a00c7}{n}}$$ `=` $$\frac{\color{#9a00c7}{n}}{2}[2\color{#e65021}{a}+(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$ $$S_{\color{#9a00c7}{20}}$$ `=` $$\frac{\color{#9a00c7}{20}}{2}[2\cdot\color{#e65021}{8}+(\color{#9a00c7}{20}-1)\color{#00880A}{4}]$$ Substitute known values `=` $$10[16+(19\cdot4)]$$ Evaluate `=` $$10(16+76)$$ `=` `10*92` `=` `920` `S_20=920` -
Question 3 of 5
3. Question
Find the sequence given that:`S_10=145``S_20=590`Hint
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Sum of an Arithmetic Sequence
$$S_{\color{#9a00c7}{n}}=\frac{\color{#9a00c7}{n}}{2}[2\color{#e65021}{a}+(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$General Rule of an Arithmetic Sequence
$$U_{\color{#9a00c7}{n}}=\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$First, use the sum formula to both sums and transform them into general rule form`S_10`$$S_{\color{#9a00c7}{n}}$$ `=` $$\frac{\color{#9a00c7}{n}}{2}[2\color{#e65021}{a}+(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$ $$S_{\color{#9a00c7}{10}}$$ `=` $$\frac{\color{#9a00c7}{10}}{2}[2\color{#e65021}{a}+(\color{#9a00c7}{10}-1)\color{#00880A}{d}]$$ Substitute known values `145``divide5` `=` `5(2a+9d)``divide5` Divide both side by `5` `29` `=` `2a+9d` `S_20`$$S_{\color{#9a00c7}{20}}$$ `=` $$\frac{\color{#9a00c7}{20}}{2}[2\color{#e65021}{a}+(\color{#9a00c7}{20}-1)\color{#00880A}{d}]$$ Substitute known values `590``divide10` `=` `10(2a+19d)``divide10` Divide both side by `10` `59` `=` `2a+19d` Next, solve for the value of `d` by subtracting the `S_10`’s general rule form from the `S_20`’s general rule form`59``-``29` `=` `(``2a+19d``)-(``2a+9d``)` `30``divide10` `=` `10d``divide10` Divide both sides by `10` `3` `=` `d` `d` `=` `3` Next, substitute `d` to one of the general rule forms to solve for `a``29` `=` `2a+9``d` `29` `=` `2a+9(``3``)` Substitute `d=5` `29` `-27` `=` `2a+27` `-27` Subtract `27` from both sides `2``divide2` `=` `2a``divide2` Divide both sides by `2` `1` `=` `a` `a` `=` `1` Finally, start with `a=1` and keep adding `d=3` to its value to get the sequence`U_1` `=` `1` `U_2` `=` `1+``3` `=` `4` `U_3` `=` `4+``3` `=` `7` `U_4` `=` `7+``3` `=` `10` `1+4+7+10…` `1+4+7+10…` -
Question 4 of 5
4. Question
Given that `S_n=301`, find the value of n`2+5+8…`- `n=` (14)
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Sum of an Arithmetic Sequence
$$S_{\color{#9a00c7}{n}}=\frac{\color{#9a00c7}{n}}{2}[2\color{#e65021}{a}+(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$Common Difference Formula
$$d=U_2-U_1=U_3-U_2$$Quadratic Formula
$$n=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$First, solve for the value of `d`.$$\color{#00880A}{d}$$ `=` $$U_2-U_1$$ `=` $$5-2$$ Substitute the first and second term `=` `3` Next, substitute the known values to the sum formula`\text(Sum of terms) [S_n]` `=` `301` `\text(First term) [a]` `=` `2` `\text(Common difference) [d]` `=` `3` $$S_{\color{#9a00c7}{n}}$$ `=` $$\frac{\color{#9a00c7}{n}}{2}[2\color{#e65021}{a}+(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$ $$301$$ `=` $$\frac{\color{#9a00c7}{n}}{2}[2\cdot\color{#e65021}{2}+(\color{#9a00c7}{n}-1)\color{#00880A}{3}]$$ Substitute known values `301``times2` `=` `n/2 [4+3n-3]``times2` Multiply both sides by `2` `602` `=` `n[1+3n]` `602` `=` `3n^2+n` Distribute `602` `-602` `=` `3n^2+n` `-602` Subtract `602` from both sides `0` `=` `3n^2+n-602` `3n^2+n-602` `=` `0` Finally, use the quadratic formula to solve for `n``a` `=` `3` `b` `=` `1` `c` `=` `-602` $$n$$ `=` $$\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$ `=` $$\frac{-1\pm \sqrt{(-1)^2-4\cdot3\cdot(-603)}}{2(3)}$$ Substitute known values `=` $$\frac{-1\pm \sqrt{1+7224}}{6}$$ Evaluate `=` $$\frac{-1\pm \sqrt{7225}}{6}$$ `=` $$\frac{-1\pm 85}{6}$$ Solve for the two values of `n`.`n` `=` $$\frac{-1+85}{6}$$ `=` $$\frac{84}{6}$$ `=` `14` `n` `=` $$\frac{-1-85}{6}$$ `=` $$\frac{-86}{6}$$ `=` $$-14.\overline{33}$$ Since the negative value has a repeating decimal, it is considered irrational.Hence, the value of `n` is `14`.`n=14` -
Question 5 of 5
5. Question
Given that `S_n=39`, find the value of the last term`1 1/2+1 3/4+2+2 1/4…`Hint
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Sum of an Arithmetic Sequence
$$S_{\color{#9a00c7}{n}}=\frac{\color{#9a00c7}{n}}{2}[\color{#e65021}{a}+\color{#00880A}{l}]$$Common Difference Formula
$$d=U_2-U_1=U_3-U_2$$General Rule of an Arithmetic Sequence
$$U_{\color{#9a00c7}{n}}=\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$First, solve for the value of `d`.$$\color{#00880A}{d}$$ `=` $$U_2-U_1$$ `=` `1 3/4-1 1/2` Substitute the first and second term `=` `1/4` Next, substitute the known values to the sum formula`\text(Sum of terms) [S_n]` `=` `39` `\text(First term) [a]` `=` `1 1/2` `\text(Common difference) [d]` `=` `1/4` $$S_{\color{#9a00c7}{n}}$$ `=` $$\frac{\color{#9a00c7}{n}}{2}[2\color{#e65021}{a}+(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$ $$39$$ `=` $$\frac{\color{#9a00c7}{n}}{2}\left[2\cdot\color{#e65021}{1\frac{1}{2}}+(\color{#9a00c7}{n}-1)\color{#00880A}{\frac{1}{4}}\right]$$ Substitute known values `39``times2` `=` `n/2 [3+n/4-1/4]``times2` Multiply both sides by `2` `78` `=` `n[11/4+n/4]` `78` `=` `(11n)/4+(n^2)/4` Distribute `78``times4` `=` `((11n)/4+(n^2)/4)``times4` Multiply both sides by `4` `312` `-312` `=` `11n+n^2` `-312` Subtract `312` from both sides `0` `=` `n^2+11n-312` `n^2+11n-312` `=` `0` Since the equation is in standard form `(ax^2+bx+c=0)` we can factorise using the cross method.`n^2` `+11``n` `-312``=0`To factorise, we need to find two numbers that add to `11` and multiply to `-312``24` and `-13` fit both conditions`24 + (-13)` `=` `11` `(24) xx (-13)` `=` `-312` 
Read across to get the factors.`(n+24)(n-13)`Since we need a positive value for `n`, we need to use `n=13`.Finally, substitute the known values to the general rule`\text(Number of terms)``[n]` `=` `13` `\text(First term)``[a]` `=` `1 1/2` `\text(Common Difference)``[d]` `=` `1/4` $$U_{\color{#9a00c7}{n}}$$ `=` $$\color{#e65021}{a}+[(\color{#9a00c7}{n}-1)\color{#00880A}{d}]$$ $$U_{\color{#9a00c7}{13}}$$ `=` $$\color{#e65021}{1\frac{1}{2}}+[(\color{#9a00c7}{13}-1)\color{#00880A}{\frac{1}{4}}]$$ Substitute known values `=` `1 1/2+12/4` Evaluate `=` `4 1/2` `\text(Last term)=4 1/2`