This was a big meta-analysis, a study of about 39 previous studies. It included only randomized control trials (RCTs):
This study conducted the most comprehensive systematic review of RCTs in adults >50 years of age to date.
Here’s the pdf, no pay wall:
Exercise Interventions For Cognitive Function In Adults Older Than 50: A Systematic Review With Meta-Analysis, British Journal of Sports Medicine, Online 30 March 2017
It found (I know you won’t be surprised):
CONCLUSIONS: This meta-analysis showed that physical exercise interventions are effective at improving the cognitive function of older adults, regardless of baseline cognitive status. Interventions of aerobic, resistance training, multicomponent training and tai chi were similarly effective. The findings suggest that an exercise programme with components of both aerobic and resistance-type training, of at least moderate intensity and at least 45 min per session, on as many days of the week as possible, is beneficial to cognitive function in adults aged >50 years.
Even a bit of tai chi a couple times a week helped, which is good to know, since it gets harder to exercise as we age (says Dr Dean Burnett, lecturer in neuroscience and psychiatry at Cardiff University):
“It could lead to increased pressure for the 50-plus age group to exercise more in order to stay mentally healthy, which is good advice but also overlooks the fact that as we age it’s increasingly difficult to engage in physical activity, as our bodies are simply less capable of it,” he said.
Look how slow she moves. That in itself is a lesson. One more thing, see how she rounds out her lower back, before and after? This is classic lower back “stretch.”
By the way … Also in that BBC article:
As well as staying physically active, Dr David Reynolds, from Alzheimer’s Research UK, said it was equally important to look after our brains by staying mentally active, eating a balanced diet, drinking only in moderation and not smoking.
Honest question … How would you describe a “balanced diet?”
Fanatic Cook 
I enjoyed seeing the Tai Chi video. Have been studying it for several years now and always like seeing the inevitable differences in performance. As to a balanced diet, it seems to me we’re all different as to how a particular diet affects us. I’ve experimented with many and am happy now with plant-based basically including nuts and plant fats such as avocado.
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You do tai chi? I admire that. It looks like it would take patience to learn.
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When I hear “eat a balanced diet” I think it refers to eating a bit from the food groups: fruit, veg, grains, dairy, eggs, meat. Or that it means eating everything in moderation. I don’t think either of those are a prescription for health. In fact, I think the phrases (“balanced diet” “in moderation”) are a cop out.
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