Step 1 — Convert the relation into a simpler form
Start with the given equation: \(S_4 = \dfrac{1}{5}\,(S_8 - S_4)\). Multiply both sides by 5: \(5S_4 = S_8 - S_4\). Bring \(S_4\) terms together: \(6S_4 = S_8\).
Step 2 — Use the formula for sum of n terms of an AP
The sum of the first \(n\) terms of an AP with first term \(a\) and common difference \(d\) is \(S_n=\dfrac{n}{2}\big(2a+(n-1)d\big)\). Apply this for \(n=4\) and \(n=8\):
\(S_4=\dfrac{4}{2}\big(2a+(4-1)d\big)=2\big(2a+3d\big)\),
\(S_8=\dfrac{8}{2}\big(2a+(8-1)d\big)=4\big(2a+7d\big).\)
Step 3 — Substitute into \(6S_4 = S_8\) and solve for \(d\)
Use \(6S_4 = S_8\):
\(6\cdot\big[2(2a+3d)\big] = 4(2a+7d).\)
Plug \(a=3\):
\(6\cdot 2\big(2\cdot 3 + 3d\big) = 4\big(2\cdot 3 + 7d\big)\) ⇒ \(12(6 + 3d) = 4(6 + 7d).\)
Simplify step-by-step:
Left: \(12\cdot 6 + 12\cdot 3d = 72 + 36d.\)
Right: \(4\cdot 6 + 4\cdot 7d = 24 + 28d.\)
Equate and isolate \(d\):
\(72 + 36d = 24 + 28d\) ⇒ \(36d - 28d = 24 - 72\) ⇒ \(8d = -48\) ⇒ \(d = -6\).
Step 4 — Quick numeric sanity check (compute S₄ and S₈ directly)
Write first few terms of the AP with \(a=3\) and \(d=-6\):
Terms: \(a_1=3,\; a_2=3-6=-3,\; a_3=-9,\; a_4=-15,\; a_5=-21,\; a_6=-27,\; a_7=-33,\; a_8=-39,\dots\)
Now compute partial sums:
\(S_4 = 3 + (-3) + (-9) + (-15) = -24.\)
\(S_8 = S_4 + (-21) + (-27) + (-33) + (-39) = -24 - 120 = -144.\)
Check relation: \(6S_4 = 6(-24) = -144 = S_8\). The relation holds — \(d=-6\) is correct.
Step 5 — Compute \(S_{20}\)
Use the sum formula with \(n=20\), \(a=3\), \(d=-6\):
\(S_{20}=\dfrac{20}{2}\big(2a+(20-1)d\big)=10\big(2\cdot 3 + 19(-6)\big).\)
Evaluate inside the bracket:
\(2\cdot 3 + 19(-6) = 6 - 114 = -108.\)
Thus \(S_{20} = 10\times (-108) = \mathbf{-1080}.\)
Interpretation
The sum \(S_{20}\) is negative because the common difference is large and negative: after the first term the sequence quickly becomes negative and the negative terms dominate the sum of the first 20 terms.
Common mistakes to avoid
• Arithmetic slip when simplifying \(12(6+3d)=4(6+7d)\).
• Forgetting to substitute \(a=3\) before expanding.
• Sign errors when computing sums of negative terms — always compute stepwise or use the formula.
Final answer
\( \boxed{S_{20} = -1080} \)
If \[ f(x) = \int \frac{1}{x^{1/4} (1 + x^{1/4})} \, dx, \quad f(0) = -6 \], then f(1) is equal to: