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Opener Kohli (70 not out) and one-down Jacks (100 not out) shared 166 runs for the unbroken second wicket to overhaul the target of 201 with 24 balls to spare.
RCB reached 206 for 1 in 16 overs.
Earlier, Gujarat Titans posted 200 for 3 after being invited to bat.
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For RCB, Swapnil Singh, Mohammed Siraj and Glenn Maxwell took a wicket apiece.
Brief Scores:
Gujarat Titans: 200 for 3 in 20 overs (Sai Sudharsan 84 not out, M Shahrukh Khan 58; Swapnil Singh 1/23).
Royal Challengers Bengaluru: 206 for 1 in 16 overs (Virat Kohli 70 not out, Will Jacks 100 not out).